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\delta^{}f\betaql#1{\betagin{equation}\lambdabel{#1}}

\title[Comultiplication rules]
{Comultiplication rules for the double
Schur\\[0.2em] functions and Cauchy identities}

\author{A. I. Molev}

\address{School of Mathematics and Statistics\\
University of Sydney, NSW 2006, Australia}
\email{alexm\hspace{0.09em}@\hspace{0.1em}maths.usyd.edu.au}

\date{} 

\betagin{abstract}
The double Schur functions form a distinguished basis
of the ring $\Lambdambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ which is a multiparameter
generalization of the ring of symmetric functions $\Lambdambda(x)$.
The canonical comultiplication on $\Lambdambda(x)$ is extended
to $\Lambdambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ in a natural way so that the double power
sums symmetric functions are primitive elements.
We calculate the dual Littlewood--Richardson coefficients
in two different ways thus
providing comultiplication rules for the
double Schur functions. We also
prove multiparameter analogues of the Cauchy identity.
A new family of Schur type functions plays the role
of a dual object in the identities.
We describe some properties of these dual
Schur functions including a combinatorial presentation
and an expansion formula in terms of the ordinary
Schur functions. The dual Littlewood--Richardson coefficients
provide a multiplication rule for the dual Schur functions.
\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}
\lambdabel{sec:int}
\setcounter{equation}{0}

The ring $\Lambda=\Lambda(x)$ of symmetric functions in the
set of variables $x=(x_1,x_2,\dots)$ admits a multiparameter
generalization $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$, where $a$ is a sequence of variables
$a=(a_i)$, $i\in\mathbb{Z}\tss$. Let $\mathbb{Q}\tss[a]$ denote the ring
of polynomials in the variables $a_i$ with rational coefficients.
The ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ is generated over $\mathbb{Q}\tss[a]$ by
the {\it double power sums symmetric functions\/}
\betaql{powess}
p_k(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{i=1}^{\infty} (x_i^k-a_i^k).
\end{equation}
Moreover, it possesses a distinguished basis
over $\mathbb{Q}\tss[a]$ formed by the
{\it double Schur functions\/} $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ parameterized
by partitions $\lambda$.
The double Schur functions $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
are closely related to the `factorial'
or `double' Schur polynomials $s_{\lambda}(x|a)$ which
were introduced by
Goulden and Greene~\cite{gg:nt} and Macdonald~\cite{m:sf}
as a generalization of the factorial Schur polynomials
of Biedenharn and Louck~\cite{bl:nc, bl:ib}.
Moreover, the polynomials $s_{\lambda}(x|a)$
are also obtained as a special case of
the double Schubert polynomials of Lascoux and
Sch\"utzenberger; see \cite{cll:fd}, \cite{l:sf}.
A formal definition of the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
and its basis elements $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
can be found in a paper of Okounkov~\cite[Remark~2.11]{o:ni}
and reproduced below in Section~\ref{sec:def}.
The ring $\Lambda$ is obtained from $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ in
the specialization $a_i=0$ for all $i\in\mathbb{Z}\tss$
while the elements $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ turn into
the classical Schur functions $s_{\lambda}(x)\in\Lambda$;
see Macdonald~\cite{m:sfh} for a detailed account
of the properties of $\Lambda$.

Another specialization $a_i=-i+1$
for all $i\in\mathbb{Z}\tss$ yields
the ring of {\it shifted symmetric functions\/}
$\Lambda^*$, introduced
and studied by Okounkov and Olshanski~\cite{oo:ss}.
Many combinatorial results of \cite{oo:ss}
can be reproduced for the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ in a
rather straightforward way.
The respective specializations of the double Schur functions
in $\Lambda^*$, known as the {\it shifted Schur functions\/}
were studied in \cite{o:qi}, \cite{oo:ss} in relation with
the higher Capelli identities
and quantum immanants for the Lie algebra $\mathfrak{gl}_n$.

In a different kind of specialization,
the double Schur functions become
the equivariant Schubert classes
on Grassmannians; see e.g.
Knutson and Tao~\cite{kt:pe},
Fulton~\cite{f:ec} and
Mihalcea~\cite{m:gf}.
The structure coefficients $c_{\lambda\mu}^{\,\nu}(a)$
of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ in the basis of
$s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$, defined by the expansion
\betaql{lrpoldef}
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\nu} c_{\lambda\mu}^{\,\nu}(a)\, s_{\,s\nu}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation}
were called the {\it Littlewood--Richardson
polynomials\/} in \cite{m:lr}.
Under the respective specializations
they describe the multiplicative structure
of the equivariant cohomology ring
on the Grassmannian and the center
of the enveloping algebra $ {\rm U}(\mathfrak{gl}_n)$.
The polynomials $c_{\lambda\mu}^{\,\nu}(a)$
possess the Graham positivity property:
they are polynomials in the differences $a_i-a_j$, $i<j$,
with positive integer coefficients; see \cite{g:pe}.
Explicit positive formulas for the polynomials $c_{\lambda\mu}^{\,\nu}(a)$
were found in \cite{kt:pe}, \cite{k:elr} and \cite{m:lr};
an earlier formula found in \cite{ms:lr} lacks
the positivity property.
The Graham positivity brings natural combinatorics
of polynomials into the structure theory of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$.
Namely, the entries of some transition
matrices between bases of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ such as
analogues of the Kostka numbers, turn out to be
Graham positive.

The {\it comultiplication\/} on the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ is
the $\mathbb{Q}\tss[a]$-linear ring homomorphism
\betan
\Delta:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\otimes^{}_{\,\mathbb{Q}\tss[a]} \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation*}
defined on the generators by
\betan
\Delta\big(p_k(x{\tss|\hspace{-1.5pt}|\tss} a)\big)= p_k(x{\tss|\hspace{-1.5pt}|\tss} a)\otimes 1+1\otimes p_k(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
In the specialization $a_i=0$ this homomorphism turns into the
comultiplication on the ring of symmetric functions $\Lambda$;
see \cite[Chapter~I]{m:sfh}.
Define the {\it dual Littlewood--Richardson
polynomials\/}
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
as the coefficients in the expansion
\betan
\Delta\big(s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)=\sum_{\lambda,\,\mu}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
\otimes s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
The central problem we address in this paper is
calculation of the polynomials $\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
in an explicit form. Note that
if $|\nu|=|\lambda|+|\mu|$ then
$c_{\lambda\mu}^{\,\nu}(a)=\widehat c_{\lambda\mu}^{\,\,s\nu}(a)
=c_{\lambda\mu}^{\,\,s\nu}$ is the
Littlewood--Richardson coefficient.
Moreover,
\betan
c_{\lambda\mu}^{\,\nu}(a)=0\quad\text{unless}
\quad |\nu|\leqslant|\lambda|+|\mu|,\quad\text{and}\quad
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=0\quad\text{unless}
\quad |\nu|\mathfrak{g}eqslant|\lambda|+|\mu|.
\end{equation*}
We will show that the polynomials
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ can be interpreted
as the multiplication coefficients for certain
analogues of the Schur functions,
\betan
\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\nu} \widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, \widehat s_{\,s\nu}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
where the $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ are symmetric
functions in $x$ which we call the {\it dual Schur functions\/}
(apparently, the term `dual double Schur functions'
would be more precise; we have chosen a shorter name
for the sake of brevity).
They can be given by the combinatorial formula
\betaql{combi}
\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{T}
\mathfrak{p}rod_{\alpha\in\lambda}X_{T(\alpha)}(a^{}_{-c(\alpha)+1},a^{}_{-c(\alpha)}),
\end{equation}
summed over the {\it reverse\/} $\lambda$-tableaux $T$, where
\betan
X_i(g,h)=
\frac{x_i\,(1-g\, x_{i-1})\dots (1-g\, x_1)}
{(1-h\, x_{i})\dots (1-h\, x_1)},
\end{equation*}
and $c(\alpha)=j-i$ denotes the content of the box $\alpha=(i,j)$;
see Section~\ref{sec:dua} below.

We calculate in an explicit form the
coefficients of the expansion of $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
as a series of the Schur functions $s_{\mu}(x)$ and vice versa.
This makes it possible to express
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ explicitly as polynomials
in the $a_i$ with the use of the Littlewood--Richardson
coefficients $c_{\lambda\mu}^{\,\,s\nu}$.

The combinatorial formula \eqref{combi}
can be used to define the skew dual Schur functions, and
we show that the following decomposition holds
\betan
\widehat s_{\,s\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\lambda} c_{\lambda\mu}^{\,\nu}(a)\, \widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
where the $c_{\lambda\mu}^{\,\nu}(a)$ are the Littlewood--Richardson
polynomials.

The functions $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ turn out to be dual
to the double Schur functions
via the following analogue of the classical Cauchy identity:
\betaql{cauchy}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation}
where $\mathcal{P}$ denotes the set of all partitions and
$y=(y_1,y_2,\dots)$ is a set of variables.

The dual Schur functions $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
are elements of the extended
ring $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ of formal series of elements of
$\Lambda(x)$ whose coefficients are polynomials in the $a_i$.
If $x=(x_1,x_2,\dots, x_n)$ is a finite set of variables
(i.e., $x_i=0$ for $i\mathfrak{g}eqslant n+1$), then $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
can be defined as the ratio
of alternants by analogy with the classical Schur polynomials.
With this definition of the dual Schur functions,
the identity \eqref{cauchy} can be deduced from
the `dual Cauchy formula' obtained in \cite[(6.17)]{m:sf}
and which is a
particular case of the Cauchy identity
for the double Schubert polynomials \cite{l:cc}.
An independent proof of a version of \eqref{cauchy}
for the shifted Schur functions (i.e., in the specialization
$a_i=-i+1$) was given by Olshanski~\cite{o:un}.
In the specialization $a_i=0$ each $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ becomes
the Schur function $s_{\lambda}(x)$, and \eqref{cauchy} turns into
the classical Cauchy identity.

We will also need a super version of the ring
of symmetric functions. The elements
\betaql{poss}
p_k(x/y)=\sum_{i=1}^{\infty}\big(x_i^k+(-1)^{k-1}y_i^k\big)
\end{equation}
with $k=1,2,\dots$ are generators of
the ring of {\it supersymmetric functions\/}
which we will regard as a $\mathbb{Q}\tss[a]$-module
and denote by $\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)$. A distinguished basis
of $\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ was introduced by
Olshanski, Regev and Vershik~\cite{orv:fs}.
In a certain specialization the basis elements
become the {\it Frobenius--Schur functions\/}
$Fs_{\lambda}$ associated with the relative dimension
function on partitions; see \cite{orv:fs}.
In order to indicate dependence on the variables,
we will denote the basis elements by $s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$
and call them the ({\it multiparameter\/})
{\it supersymmetric Schur functions\/}.
They
are closely related to the {\it factorial
supersymmetric Schur polynomials\/} introduced in \cite{m:fs};
see Section~\ref{sec:def} for precise formulas.
Note that the
evaluation map $y_i\mapsto -a_i$ for all $i\mathfrak{g}eqslant 1$
defines an isomorphism
\betaql{isom}
\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)\to \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
The images of the generators \eqref{poss}
under this isomorphism
are the
double power sums symmetric functions
\eqref{powess}.
We will show that under the isomorphism \eqref{isom} we have
\betaql{isomimsf}
s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\mapsto s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
Due to \cite{orv:fs}, the
supersymmetric Schur functions possess a remarkable
combinatorial presentation in terms of diagonal-strict or
`shuffle' tableaux.
The isomorphism \eqref{isom}
implies the corresponding
combinatorial presentation for $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
and allows us to introduce the {\it skew
double Schur functions\/}
$s_{\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$.
The dual Littlewood--Richardson
polynomials
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ can then be
found from the expansion
\betaql{skewde}
s_{\,s\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation}
which leads to an alternative rule for the calculation of
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$. This rule relies on
the combinatorial objects called `barred tableaux'
which were introduced in \cite{ms:lr}
for the calculation of the
polynomials $c_{\lambda\mu}^{\,s\nu}(a)$; see also
\cite{k:elr}, \cite{k:pf} and \cite{m:lr}.

The coefficients in the expansion of $s_{\mu}(x)$
in terms of the $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ turn out to coincide with
those in the decomposition of
$s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ in terms of the ordinary supersymmetric
Schur functions $s_{\lambda}(x/y)$ thus providing another
expression for these coefficients; cf. \cite{orv:fs}.

The identity \eqref{cauchy} allows us to introduce a pairing
between the rings $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ so that
the respective families
$\{s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\}$ and $\{\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\}$
are dual to each other. This leads to a natural definition
of the monomial and forgotten symmetric
functions in $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
by analogy with \cite{m:sfh} and provides a relationship
between the transition matrices relating different
bases of these rings.

It is well known that
the ring of symmetric functions $\Lambda$ admits an involutive
automorphism $\omega^{}:\Lambda\to\Lambda$ which interchanges the elementary
and complete symmetric functions; see \cite{m:sfh}.
We show that there is an isomorphism
$\omega^{}_a:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$, and
$\omega^{}_a$ has the property $\omega^{}_{a'}\circ\omega^{}_{a}=\text{id}$,
where $a'$ denotes the sequence of parameters with $(a')_i=-a_{-i+1}$.
Moreover, the images of the natural bases elements of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
with respect to $\omega^{}_a$
can be explicitly described; see also \cite{oo:ss}
where such an involution was
constructed for the specialization $a_i=-i+1$, and
\cite{orv:fs} for its super version.
Furthermore, using a symmetry property of the
supersymmetric Schur functions,
we derive the symmetry properties of the
Littlewood--Richardson polynomials and their dual counterparts
\betan
c_{\lambda\mu}^{\,s\nu}(a)=c_{\lambda'\mu'}^{\,s\nu^{\,s\mathfrak{p}rime}}(a')
\qquad\text{and}\qquad
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=
\widehat c_{\lambda'\mu'}^{\,\,s\nu^{\,s\mathfrak{p}rime}}(a'),
\end{equation*}
where $\rho^{\,s\mathfrak{p}rime}$ denotes the conjugate
partition to any partition $\rho$.
In the context of equivariant cohomology,
the first relation is a consequence of the Grassmann duality;
see e.g. \cite[Lecture~8]{f:ec} and \cite{kt:pe}.

An essential
role in the proof of \eqref{cauchy} is played by
interpolation formulas for symmetric functions. The interpolation
approach goes back to the work of Okounkov~\cite{o:qi, o:ni},
where the key {\it vanishing theorem\/} for the double Schur functions
$s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ was proved; see also \cite{oo:ss}.
In a more general context, the Newton interpolation
for polynomials in several variables relies on the theory
of Schubert polynomials of Lascoux and
Sch\"utzenberger; see \cite{l:sf}.
The interpolation approach leads to a recurrence
relation for the coefficients $c_{P,\,\mu}^{\,\nu}(a)$
in the expansion
\betaql{interp}
P\, s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\nu}
c_{P,\,\mu}^{\,\nu}(a)\, s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a),\qquad P\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation}
as well as to an explicit formula for the $c_{P,\,\mu}^{\,\nu}(a)$
in terms of the values of $P$; see \cite{ms:lr}.
Therefore, the (dual) Littlewood--Richardson polynomials
and the entries of the transition matrices between
various bases of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ can be given as rational functions
in the variables $a_i$.
Under appropriate specializations,
these formulas imply some
combinatorial identities
involving Kostka numbers, irreducible
characters of the symmetric
group and dimensions of skew diagrams; cf. \cite{oo:ss}.

I am grateful to Grigori Olshanski for valuable
remarks and discussions.

\section{Double and supersymmetric Schur functions}
\lambdabel{sec:def}
\setcounter{equation}{0}

\subsection{Definitions and preliminaries}
\lambdabel{subsec:dpr}

Recall the definition of the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ from
\cite[Remark~2.11]{o:ni};
see also \cite{m:lr}.
For each nonnegative integer $n$
denote by $\Lambda_n$ the ring of symmetric
polynomials in $x_1,\dots,x_n$ with coefficients in $\mathbb{Q}\tss[a]$
and let $\Lambda^k_n$ denote the $\mathbb{Q}\tss[a]$-submodule of $\Lambda_n$
which consists of
the polynomials $P_n(x_1,\dots,x_n)$ such that the total degree
of $P_n$ in the variables $x_i$ does not exceed $k$.
Consider the evaluation maps
\betaql{eval}
\varphi_n:\Lambda^{k}_{n}\to\Lambda^{k}_{n-1},\qquad
P_n(x_1,\dots,x_n)\mapsto
P_n(x_1,\dots,x_{n-1},a_n)
\end{equation}
and the corresponding inverse limit
\betan
\Lambda^{k}=\lim_{\longleftarrow}
\Lambda^{k}_n,\qquad n\to\infty.
\end{equation*}
The elements of $\Lambda^{k}$ are sequences
$P=(P_0,P_1,P_2,\dots)$ with $P_n\in \Lambda^{k}_n$
such that
\betan
\varphi_n(P_n)=P_{n-1}
\qquad\text{for}\quad n=1,2,\dots.
\end{equation*}
Then the union
\betan
\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)= \bigcup_{k\mathfrak{g}eqslant 0}\Lambda^{k}
\end{equation*}
is a ring with the product
\betan
P\,s Q=(P_0\,s Q_0,P_1\,s Q_1,P_2\,s Q_2,\dots),
\qquad Q=(Q_0,Q_1,Q_2,\dots).
\end{equation*}
The elements of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ may be regarded
as formal series in the variables
$x_i$ with coefficients in $\mathbb{Q}\tss[a]$.
For instance, the sequence of polynomials
\betan
\sum_{i=1}^n (x_i^k-a_i^k),\qquad n \mathfrak{g}eqslant 0,
\end{equation*}
determines the {\it double power sums symmetric function\/}
\eqref{powess}.

Note that if $k$ is fixed, then the evaluation
maps \eqref{eval} are isomorphisms for
all sufficiently large values of $n$. This allows one
to establish many properties of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
by working with finite sets of variables $x=(x_1,\dots,x_n)$.

Now we recall the definition and some key properties of the
double Schur functions. We basically follow
\cite[6th~Variation]{m:sf} and \cite{o:ni},
although our notation is slightly different.
A partition
$\lambda$ is a weakly decreasing sequence
$\lambda=(\lambda_1,\dots,\lambda_l)$ of integers $\lambda_i$ such that
$\lambda_1\mathfrak{g}eqslant\dots\mathfrak{g}eqslant\lambda_l\mathfrak{g}eqslant 0$.
Sometimes this sequence
is considered to be completed by a finite or infinite
sequence of zeros.
We will identify $\lambda$ with its diagram
represented graphically as
the array of left justified rows of unit boxes
with $\lambda_1$ boxes in the top row, $\lambda_2$ boxes in the second
row, etc. The total number of boxes in $\lambda$
will be denoted by $|\lambda|$ and the number of nonzero rows
will be called the length of $\lambda$ and denoted
$\ell(\lambda)$.
The transposed diagram
$\lambda'=(\lambda'_1,\dots,\lambda'_p)$ is obtained from $\lambda$
by applying the symmetry
with respect to the main diagonal, so that
$\lambda'_j$ is the number of boxes in the $j$-th column of $\lambda$.
If $\mu$ is a diagram contained in $\lambda$, then the
skew diagram $\lambda/\mu$ is the set-theoretical difference
of diagrams $\lambda$ and $\mu$.

Suppose now that $x=(x_1,\dots,x_n)$ is a finite set
of variables. For any $n$-tuple of nonnegative integers
$\alpha=(\alpha_1,\dots,\alpha_n)$ set
\betan
A_{\alpha}(x{\tss|\hspace{-1.5pt}|\tss} a)=\delta^{}t\big[(x_i{\tss|\hspace{-1.5pt}|\tss} a)^{\alpha_j}\big]_{i,j=1}^n,
\end{equation*}
where $(x_i{\tss|\hspace{-1.5pt}|\tss} a)^0=1$ and
\betan
(x_i{\tss|\hspace{-1.5pt}|\tss} a)^r=(x_i-a_n)(x_i-a_{n-1})\dots (x_i-a_{n-r+1}),
\qquad r\mathfrak{g}eqslant 1.
\end{equation*}
For any partition $\lambda=(\lambda_1,\dots,\lambda_n)$ of length
not exceeding $n$
set
\betan
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\frac{A_{\lambda+\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)}{A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)},
\end{equation*}
where $\delta^{}=(n-1,\dots,1,0)$. Note that since $A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)$
is a skew-symmetric polynomial in $x$ of degree $n(n-1)/2$, it
coincides with the Vandermonde determinant,
\betan
A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)=\mathfrak{p}rod_{1\leqslant i<j\leqslant n}(x_i-x_j)
\end{equation*}
and so $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ belongs to the ring $\Lambda_n$.
Moreover,
\betan
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{\lambda}(x)+\ \text{lower degree terms in}\ \  x,
\end{equation*}
where $s_{\lambda}(x)$ is the Schur polynomial; see e.g.
\cite[Chapter~I]{m:sfh}.
We also set $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=0$ if $\ell(\lambda)> n$.
Then under the evaluation map \eqref{eval} we have
\betan
\varphi_n: s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto s_{\lambda}(x'{\tss|\hspace{-1.5pt}|\tss} a),
\qquad x'=(x_1,\dots,x_{n-1}),
\end{equation*}
so that the sequence $\big(s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\ |\ n\mathfrak{g}eqslant 0\big)$
defines an element of the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$. We will keep
the notation $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ for this element of
$\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$, where $x$ is now understood as the infinite sequence of
variables, and call it the {\it double Schur function\/}.

By a {\it reverse $\lambda$-tableau\/}
$T$ we will mean a tableau obtained by filling in the boxes
of $\lambda$ with the positive integers in such a way that
the entries weakly decrease along the rows and strictly decrease
down the columns. If $\alpha=(i,j)$ is a box of $\lambda$
in row $i$ and column $j$,
we let $T(\alpha)=T(i,j)$ denote the entry of $T$ in the box $\alpha$
and let $c(\alphapha)=j-i$ denote the content of
this box.
The double Schur functions admit the following
tableau presentation
\betaql{defdouble}
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{T}
\mathfrak{p}rod_{\alpha\in\lambda}
(x^{}_{T(\alphapha)}-a^{}_{T(\alphapha)-c(\alpha)}),
\end{equation}
summed over all reverse $\lambda$-tableaux $T$.

When the entries of $T$ are restricted to
the set $\{1,\dots,n\}$, formula \eqref{defdouble}
provides the respective tableau presentation of
the polynomials $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ with $x=(x_1,\dots,x_n)$.
Moreover, in this case the formula can be extended to
skew diagrams and we define the corresponding
polynomials by
\betaql{defdoubleskew}
\widetilde s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{T}
\mathfrak{p}rod_{\alpha\in\theta}
(x^{}_{T(\alphapha)}-a^{}_{T(\alphapha)-c(\alpha)}),
\end{equation}
summed over all reverse $\theta$-tableaux $T$
with entries in $\{1,\dots,n\}$, where $\theta$
is a skew diagram. We suppose that $\widetilde s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=0$
unless all columns of $\theta$ contain at most $n$ boxes.

\begin{remark}\lambdabel{rem:skew}
(i)\quad
Although the polynomials \eqref{defdoubleskew}
belong to the ring $\Lambda_n$, they are generally
not consistent with respect to
the evaluation maps \eqref{eval}.
We used different notation in \eqref{defdouble} and
\eqref{defdoubleskew} in order to distinguish between
the polynomials $\widetilde s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and the
skew double Schur functions $s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)$
to be introduced in
Definition~\ref{def:skdsf} below.

\noindent
(ii)\quad
In order to relate our notation to \cite{m:sf}, note that
for the polynomials $\widetilde s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)$ with $x=(x_1,\dots,x_n)$
we have
\betan
\widetilde s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{\theta}(x\,s|\,s u),
\end{equation*}
where the sequences $a=(a_i)$ and $u=(u_i)$ are related by
\betaql{seqs}
u_i=a_{n-i+1},\qquad i\in\mathbb{Z}\tss.
\end{equation}
The polynomials $s_{\theta}(x\,s|\,s u)$ are often called
the {\it factorial Schur polynomials\/} ({\it functions\/})
in the literature. They can be given by
the combinatorial formula
\betaql{facsf}
s_{\theta}(x\,s|\,s u)=\sum_{T}
\mathfrak{p}rod_{\alpha\in\theta}
(x^{}_{T(\alphapha)}-u^{}_{T(\alphapha)+c(\alpha)}),
\end{equation}
summed over all {\it semistandard\/} $\theta$-tableaux $T$
with entries in $\{1,\dots,n\}$; the entries of $T$
weakly increase along the rows and strictly increase
down the columns.

\noindent
(iii)\quad
If we replace $a_i$ with $c_{-i}$
and index the variables $x$ with nonnegative integers,
the double Schur functions $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ will become
the corresponding symmetric
functions of \cite{o:ni}; cf. formula (3.7) in that paper.
Moreover, under the specialization $a_i=-i+1$ for all $i\in\mathbb{Z}\tss$
the double Schur functions become the {\it shifted
Schur functions\/} of \cite{oo:ss} in the variables $y_i=x_i+i-1$.
\qed
\end{remark}

\subsection{Analogues of classical bases}

The {\it double elementary\/} and {\it complete
symmetric functions} are defined respectively by
\betan
e_k(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{(1^k)}(x{\tss|\hspace{-1.5pt}|\tss} a),\qquad h_k(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{(k)}(x{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation*}
and hence, they can be given by the formulas
\betan
\begin{aligned}
e_k(x{\tss|\hspace{-1.5pt}|\tss} a)&=\sum_{i_1>\dots>i_k}(x_{i_1}-a_{i_1})\dots
(x_{i_k}-a_{i_k+k-1}),\\
h_k(x{\tss|\hspace{-1.5pt}|\tss} a)&=\sum_{i_1\mathfrak{g}eqslant\dots\mathfrak{g}eqslant i_k}
(x_{i_1}-a_{i_1})\dots
(x_{i_k}-a_{i_k-k+1}).
\end{aligned}
\end{equation*}
Their generating functions can be written
by analogy with the classical case as in \cite{m:sfh}
and they take the form
\betagin{align}\lambdabel{gene}
1+\sum_{k=1}^{\infty} \frac{e_k(x{\tss|\hspace{-1.5pt}|\tss} a)\, t^k}
{(1+a_1\,s t)\dots(1+a_k\,s t)}&=
\mathfrak{p}rod_{i=1}^{\infty}\frac{1+x_i\,s t}{1+a_i\,s t},\\[1em]
\lambdabel{genh}
1+\sum_{k=1}^{\infty} \frac{h_k(x{\tss|\hspace{-1.5pt}|\tss} a)\, t^k}
{(1-a_0\,s t)\dots(1-a_{-k+1}\,s t)}&=
\mathfrak{p}rod_{i=1}^{\infty}\frac{1-a_i\,s t}{1-x_i\,s t};
\end{align}
see e.g. \cite{m:sf}, \cite{oo:ss}.

Given a partition
$\lambda=(\lambda_1,\dots,\lambda_l)$, set
\betan
\begin{aligned}
p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)&=p_{\lambda_1}(x{\tss|\hspace{-1.5pt}|\tss} a)\dots p_{\lambda_l}(x{\tss|\hspace{-1.5pt}|\tss} a),\\
e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)&=e_{\lambda_1}(x{\tss|\hspace{-1.5pt}|\tss} a)\dots e_{\lambda_l}(x{\tss|\hspace{-1.5pt}|\tss} a),\\
h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)&=h_{\lambda_1}(x{\tss|\hspace{-1.5pt}|\tss} a)\dots h_{\lambda_l}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{aligned}
\end{equation*}

The following proposition is easy to deduce from
the properties of the classical symmetric functions;
see \cite{m:sfh}.

\begin{prop}\lambdabel{prop:basis}
Each of the families $p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$,
$e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$, $h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
and $s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$, parameterized by all
partitions $\lambda$,
forms a basis of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ over $\mathbb{Q}\tss[a]$.
\qed
\end{prop}

In particular, each of the families
$p_k(x{\tss|\hspace{-1.5pt}|\tss} a)$,
$e_k(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $h_k(x{\tss|\hspace{-1.5pt}|\tss} a)$ with $k\mathfrak{g}eqslant 1$
is a set of algebraically independent generators of
$\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ over $\mathbb{Q}\tss[a]$.
Under the specialization $a_i=0$, the bases
of Proposition~\ref{prop:basis} turn into
the classical bases $p_{\lambda}(x)$,
$e_{\lambda}(x)$, $h_{\lambda}(x)$
and $s_{\lambda}(x)$ of $\Lambda$.
The ring of symmetric functions $\Lambda$
possesses two more bases $m_{\lambda}(x)$ and $f_{\lambda}(x)$;
see \cite[Chapter~I]{m:sfh}.
The {\it monomial
symmetric functions\/} $m_{\lambda}(x)$ are defined by
\betan
m_{\lambda}(x)=\sum_{\sigma} x_{\sigma(1)}^{\lambda_1}x_{\sigma(2)}^{\lambda_2}\dots
x_{\sigma(l)}^{\lambda_l},
\end{equation*}
summed over permutations $\sigma$ of the $x_i$
which give distinct monomials. The basis elements $f_{\lambda}(x)$
are called the {\it forgotten symmetric functions\/}, they are
defined as the images of the $m_{\lambda}(x)$ under the involution
$\omega^{}:\Lambda\to\Lambda$ which takes $e_{\lambda}(x)$ to $h_{\lambda}(x)$;
see \cite{m:sfh}.
The corresponding basis elements
$m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $f_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
in $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ will be defined in Section~\ref{sec:ome}.

\subsection{Duality isomorphism}

Introduce the sequence of variables $a'$ which is
related to the sequence $a$ by the rule
\betan
(a')_i=-a_{-i+1},\qquad i\in\mathbb{Z}\tss.
\end{equation*}
The operation $a\mapsto a'$ is clearly involutive so that
$(a')'=a$. Note that any element of
the polynomial ring $\mathbb{Q}\tss[a']$ can be identified with
the element of $\mathbb{Q}\tss[a]$ obtained by replacing each $(a')_i$
by $-a_{-i+1}$.
Define the ring homomorphism
\betan
\omega^{}_a:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')
\end{equation*}
as the $\mathbb{Q}\tss[a]$-linear map
such that
\betaql{omega}
\omega^{}_a: e_k(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto h_k(x{\tss|\hspace{-1.5pt}|\tss} a'),\qquad k=1,2,\dots.
\end{equation}
An arbitrary element of $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ can be written
as a unique linear combination of the basis elements $e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
with coefficients in $\mathbb{Q}\tss[a]$. The image of such
a linear combination under $\omega^{}_a$ is then found by
\betan
\omega^{}_a:\sum_{\lambda}c_{\lambda}(a)\, e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
\mapsto \sum_{\lambda}c_{\lambda}(a)\, h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a'),\qquad
c_{\lambda}(a)\in\mathbb{Q}\tss[a],
\end{equation*}
and $c_{\lambda}(a)$ is regarded as an element of $\mathbb{Q}\tss[a']$.
Clearly, $\omega^{}_a$ is a ring isomorphism,
since the $h_k(x{\tss|\hspace{-1.5pt}|\tss} a')$ are
algebraically independent generators of
$\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$ over $\mathbb{Q}\tss[a']$. In the case of finite set
of variables $x=(x_1,\dots,x_n)$ the respective
isomorphism $\omega^{}_a$ is defined by the same rule \eqref{omega}
with the values $k=1,\dots,n$.

\begin{prop}\lambdabel{prop:om}
We have
$\omega^{}_{a'}\circ \omega^{}_a={\rm id}^{}_{\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)}$ and
\betaql{omhla}
\omega^{}_a: h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a').
\end{equation}
\end{prop}

\begin{proof}
Relations \eqref{gene} and \eqref{genh} imply that
\betan
\Bigg(\sum_{k=0}^{\infty} \frac{(-1)^k\, e_k(x{\tss|\hspace{-1.5pt}|\tss} a)\, t^k}
{(1-a_1\,s t)\dots(1-a_k\,s t)}\Bigg)
\Bigg(\sum_{r=0}^{\infty} \frac{h_r(x{\tss|\hspace{-1.5pt}|\tss} a)\, t^r}
{(1-a_0\,s t)\dots(1-a_{-r+1}\,s t)}\Bigg)=1.
\end{equation*}
Applying the isomorphism $\omega^{}_a$, we get
\betan
\Bigg(\sum_{k=0}^{\infty} \frac{(-1)^k\, h_k(x{\tss|\hspace{-1.5pt}|\tss} a')\, t^k}
{(1+(a')_0\,s t)\dots(1+(a')_{-k+1}\,s t)}\Bigg)
\Bigg(\sum_{r=0}^{\infty} \frac{\omega^{}_a\big(h_r(x{\tss|\hspace{-1.5pt}|\tss} a)\big)\, t^r}
{(1+(a')_1\,s t)\dots(1+(a')_r\,s t)}\Bigg)=1.
\end{equation*}
Replacing here $t$ by $-t$ and comparing with the previous
identity, we can conclude that
$\omega^{}_a\big(h_r(x{\tss|\hspace{-1.5pt}|\tss} a)\big)=e_r(x{\tss|\hspace{-1.5pt}|\tss} a')$.
This proves \eqref{omhla} and the first part
of the proposition, because
$\omega^{}_{a'}\big(h_r(x{\tss|\hspace{-1.5pt}|\tss} a')\big)=e_r(x{\tss|\hspace{-1.5pt}|\tss} a)$.
\end{proof}

We will often use the shift operator
$\tau$ whose
powers act on sequences by the rule
\betan
(\tau^k a)_i=a_{k+i}\qquad\text{for}\quad k\in\mathbb{Z}\tss.
\end{equation*}
The following analogues of the
Jacobi--Trudi
and N\"{a}gelsbach--Kostka formulas
are immediate from \cite[(6.7)]{m:sf}.
Namely, if the set of variables $x=(x_1,\dots,x_n)$ is
finite and $\lambda$
is a partition of length not exceeding $n$, then
\betaql{jt}
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\delta^{}t\big[h_{\lambda_i-i+j}(x{\tss|\hspace{-1.5pt}|\tss} \tau^{\,s j-1}\,s a)\big]
\end{equation}
and
\betaql{nk}
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\delta^{}t\big[e_{\lambda'_i-i+j}(x{\tss|\hspace{-1.5pt}|\tss} \tau^{\,s -j+1}\,s a)\big],
\end{equation}
where the determinants are taken over the respective sets of indices
$i,j=1,\dots,\ell(\lambda)$ and
$i,j=1,\dots,\ell(\lambda')$.

\subsection{Skew double Schur functions}

Consider now the ring of supersymmetric functions
$\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ defined in the Introduction.
Taking two finite sets of variables $x=(x_1,\dots,x_n)$
and $y=(y_1,\dots,y_n)$, define the {\it supersymmetric
Schur polynomial\/} $s_{\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$
associated with a skew diagram $\nu/\mu$ by the formula
\betaql{defssf}
s_{\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu\subseteq\,\rho\,\subseteq\,s\nu}
\widetilde s_{\nu/\rho}(x{\tss|\hspace{-1.5pt}|\tss} a)\,
s_{\rho^{\,s\mathfrak{p}rime}/\mu^{\,s\mathfrak{p}rime}}(y \,s|\,s {-}a),
\end{equation}
where the polynomials $\widetilde s_{\nu/\rho}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and
$s_{\rho^{\,s\mathfrak{p}rime}/\mu^{\,s\mathfrak{p}rime}}(y \,s|\,s{-}a)$ are defined
by the respective combinatorial formulas
\eqref{defdoubleskew} and \eqref{facsf}.
The polynomials \eqref{defssf} coincide with
the factorial supersymmetric Schur polynomials
$s_{\nu/\mu}(x/y\,s|\,s u)$
of \cite{m:fs} associated with the sequence
$u$ related to $a$ by \eqref{seqs}.
It was observed in \cite{orv:fs}
that the sequence of polynomials
$\big(s_{\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\,|\, n\mathfrak{g}eqslant 1\big)$ is consistent
with respect to the evaluations $x_n=y_n=0$ and hence,
it defines the {\it supersymmetric
Schur function\/} $s_{\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$, where
$x$ and $y$ are infinite sequences of variables
(in fact, Proposition~3.4 in \cite{orv:fs} needs to be
extended to skew diagrams which is immediate).
Moreover, in \cite{orv:fs}
these functions were given by new
combinatorial formulas.
In order to write them down,
consider the ordered alphabet
\betan
\mathbb{A}\tss=\{1'<1<2^{\,s\mathfrak{p}rime}<2<\dots\}.
\end{equation*}
Given a skew diagram $\theta$, an
$\mathbb{A}\tss$-{\it tableau\/} $T$ of shape $\theta$
is obtained by filling in the boxes of $\theta$ with
the elements of $\mathbb{A}\tss$ in such a way that the
entries of $T$ weakly increase along each row and
down each column, and for each $i=1,2,\dots$ there is at most
one symbol $i'$ in each row and at most
one symbol $i$ in each column of $T$.
The following formula gives
the supersymmetric
Schur function $s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ associated with $\theta$:
\betaql{tabapr}
s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{T}
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, unprimed}}{\alphapha\in\theta}}
\big(x^{}_{T(\alpha)}-a_{-c(\alpha)+1}\big)
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, primed}}{\alphapha\in\theta}}
\big(y^{}_{T(\alpha)}+a_{-c(\alpha)+1}\big),
\end{equation}
summed over all $\mathbb{A}\tss$-tableaux $T$ of shape $\theta$,
where the subscripts of the variables $y_i$ are identified
with the primed indices. An alternative formula is obtained
by using a different ordering of the alphabet:
\betan
\mathbb{A}\tss'=\{1<1'<2<2^{\,s\mathfrak{p}rime}<\dots\}.
\end{equation*}
The $\mathbb{A}\tss'$-tableaux $T$ of shape $\theta$ are defined
in exactly the same way as the $\mathbb{A}\tss$-tableaux, only taking
into account the new ordering. Then
\betaql{taba}
s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{T}
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, unprimed}}{\alphapha\in\theta}}
\big(x^{}_{T(\alpha)}-a_{-c(\alpha)}\big)
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, primed}}{\alphapha\in\theta}}
\big(y^{}_{T(\alpha)}+a_{-c(\alpha)}\big),
\end{equation}
summed over all $\mathbb{A}\tss'$-tableaux $T$ of shape $\theta$.

The supersymmetric
Schur functions have the following symmetry property
\betaql{symprsu}
s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=s_{\theta^{\,s\mathfrak{p}rime}}(y/x{\tss|\hspace{-1.5pt}|\tss} a')
\end{equation}
implied by their combinatorial presentation.
Moreover, if $x_i=y_i=0$ for all $i\mathfrak{g}eqslant n+1$, then
only tableaux $T$ with entries in $\{1,1',\dots,n,n'\}$
make nonzero contributions in
either \eqref{tabapr} or \eqref{taba}.

\begin{remark}\lambdabel{rem:deforv}
The supersymmetric
Schur function
$s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ given in \eqref{tabapr}
coincides with $\Sigma_{\theta;-a'}(x;y)$ as defined
in \cite[Proposition~4.4]{orv:fs}.
In order to derive \eqref{taba}, first use \eqref{symprsu},
then apply the transposition of the tableaux
with respect to the main diagonal and swap $i$ and $i^{\,s\mathfrak{p}rime}$
for each $i$.
Note that \cite{orv:fs} also contains an
equivalent combinatorial formula for $\Sigma_{\theta;a}(x;y)$
in terms of skew hooks.
\qed
\end{remark}

\begin{prop}\lambdabel{prop:imisom}
The image of the supersymmetric
Schur function $s_{\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ associated with
a {\rm(}nonskew{\rm)} diagram $\nu$ under the isomorphism \eqref{isom}
coincides with the double Schur function
$s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)$; that is,
\betan
s_{\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\big|^{}_{y=-a}=s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
where $y=-a$ denotes the evaluation $y_i=-a_i$ for $i\mathfrak{g}eqslant 1$.
\end{prop}

\begin{proof}
We may assume that the sets of variables $x$ and $y$
are finite,
$x=(x_1,\dots,x_n)$
and $y=(y_1,\dots,y_n)$. The claim
now follows from relation \eqref{defssf} with $\mu=\varnothing$,
if we observe that
$s_{\rho^{\,s\mathfrak{p}rime}}(y \,s|\,s {-}a)\big|^{}_{y=-a}=0$
unless $\rho=\varnothing$.
\end{proof}

The symmetry property \eqref{symprsu} implies
the following dual version of Proposition~\ref{prop:imisom}.

\begin{cor}\lambdabel{cor:dueva}
Under the isomorphism $\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)\to \Lambda(y{\tss|\hspace{-1.5pt}|\tss} a')$
defined by the evaluation $x_i=-(a')_i$ for all $i\mathfrak{g}eqslant 1$
we have
\betan
s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\big|^{}_{x=-a'}=s_{\theta'}(y{\tss|\hspace{-1.5pt}|\tss} a').
\end{equation*}
\end{cor}

Using Proposition~\ref{prop:imisom}, we can find the images
of the double Schur functions with respect to
the duality isomorphism $\omega^{}_a$ defined in \eqref{omega}.

\begin{cor}\lambdabel{cor:imsuf}
Under the isomorphism $\omega^{}_a:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$
we have
\betaql{omsla}
\omega^{}_a: s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto s_{\lambda'}(x{\tss|\hspace{-1.5pt}|\tss} a').
\end{equation}
\end{cor}

\begin{proof}
The Littlewood--Richardson
polynomials $c_{\lambda\mu}^{\,s\nu}(a)$ are defined
by the expansion \eqref{lrpoldef}. Hence,
by Proposition~\ref{prop:imisom}
we have
\betan
s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\, s_{\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\nu} c_{\lambda\mu}^{\,\nu}(a)\, s_{\,s\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
Using \eqref{symprsu}, we get
\betaql{symlr}
c_{\lambda\mu}^{\,s\nu}(a)=c_{\lambda'\mu'}^{\,s\nu^{\,s\mathfrak{p}rime}}(a').
\end{equation}
Now, observe that relation \eqref{omsla} can be taken as
a definition of the $\mathbb{Q}\tss[a]$-module isomorphism
$\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$. Moreover, this definition
agrees with \eqref{omega}.
Therefore, it is sufficient to verify that this
$\mathbb{Q}\tss[a]$-module isomorphism is a ring homomorphism.
Applying \eqref{symlr} we obtain
\betan
\begin{aligned}
\omega^{}_a\big(s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)
{}&{}=\sum_{\nu} c_{\lambda\mu}^{\,\nu}(a)\,
\omega^{}_a\big(s_{\,s\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)
=\sum_{\nu} c_{\lambda'\mu'}^{\,\nu^{\,s\mathfrak{p}rime}}(a')\,
s_{\,s\nu^{\,s\mathfrak{p}rime}}(x{\tss|\hspace{-1.5pt}|\tss} a')\\
{}&{}=s_{\lambda'}(x{\tss|\hspace{-1.5pt}|\tss} a')\, s_{\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a')=
\omega^{}_a\big(s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)\, \omega^{}_a\big(s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big).
\end{aligned}
\end{equation*}
\end{proof}

Proposition~\ref{prop:imisom} leads to the following definition.

\begin{defin}\lambdabel{def:skdsf}
For any skew diagram $\theta$ define the
{\it skew double Schur function\/} $s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
as the image of $s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\in\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)$
under the isomorphism \eqref{isom}; that is,
\betan
s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{\theta}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\big|^{}_{y=-a}.
\end{equation*}
Equivalently, using \eqref{tabapr} and \eqref{taba},
respectively, we have
\betaql{tabadsdu}
s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{T}
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, unprimed}}{\alphapha\in\theta}}
\big(x^{}_{T(\alpha)}-a_{-c(\alpha)+1}\big)
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, primed}}{\alphapha\in\theta}}
\big(a_{-c(\alpha)+1}-a^{}_{T(\alpha)}\big),
\end{equation}
summed over all $\mathbb{A}\tss$-tableaux $T$ of shape $\theta$; and
\betaql{tabads}
s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{T}
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, unprimed}}{\alphapha\in\theta}}
\big(x^{}_{T(\alpha)}-a_{-c(\alpha)}\big)
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, primed}}{\alphapha\in\theta}}
\big(a_{-c(\alpha)}-a^{}_{T(\alpha)}\big),
\end{equation}
summed over all $\mathbb{A}\tss'$-tableaux $T$ of shape $\theta$.
Furthermore, by \eqref{defssf} the
skew double Schur function $s_{\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$
can also be defined as the sequence of polynomials
\betaql{seqpol}
s_{\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu\subseteq\,\rho\,\subseteq\,s\nu}
\widetilde s_{\nu/\rho}(x{\tss|\hspace{-1.5pt}|\tss} a)\,
s_{\rho^{\,s\mathfrak{p}rime}/\mu^{\,s\mathfrak{p}rime}}(-a^{(n)} \,s|\,s {-}a),
\qquad n=1,2,\dots,
\end{equation}
where $x=(x_1,\dots,x_n)$ and $a^{(n)}=(a_1,\dots,a_n)$.
\qed
\end{defin}

For any partition $\mu$ introduce the sequence $a_{\mu}$
and the series $|a_{\mu}|$ by
\betan
a_{\mu}=(a_{1-\mu_1},a_{2-\mu_2},\dots)\qquad\text{and}\qquad
|a_{\mu}|=a_{1-\mu_1}+a_{2-\mu_2}+\dots.
\end{equation*}
Given any element $P(x)\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$, the value
$P(a_{\mu})$
is a well-defined element of $\mathbb{Q}\tss[a]$.
The vanishing theorem of Okounkov~\cite{o:qi, o:ni}
states that
\betan
s_{\lambda}(a_{\rho}{\tss|\hspace{-1.5pt}|\tss} a)=0\qquad\text{unless}
\quad \lambda\subseteq\rho,
\end{equation*}
and
\betaql{hoo}
s_{\lambda}(a_{\lambda}{\tss|\hspace{-1.5pt}|\tss} a)=\mathfrak{p}rod_{(i,j)\in\lambda}
\big(a^{}_{i-\lambda_i}-a^{}_{\lambda'_j-j+1}\big).
\end{equation}
This theorem can be used to derive the
interpolation formulas given in the next
proposition. In a slightly different situation this derivation
was performed in \cite[Propositions~3.3 \& 3.4]{ms:lr}
relying on the approach of \cite{oo:ss},
and an obvious
modification of those arguments works in the present context;
see also \cite{f:ec}, \cite{kt:pe}.
The expressions like $|a_{\nu}|-|a_{\mu}|$ used
below are understood
as the polynomials $\sum_{i\mathfrak{g}eqslant 1}
(a_{i-\nu_i}-a_{i-\mu_i})$.
We will write $\rho\to\sigmagma$ if the diagram $\sigma$
is obtained from the diagram $\rho$
by adding one box.

\begin{prop}\lambdabel{prop:interp}
Given an element $P(x)\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$, define the
polynomials $c_{P,\,\mu}^{\,\nu}(a)$
by the expansion
\betaql{interpo}
P(x)\, s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\nu}
c_{P,\,\mu}^{\,\nu}(a)\, s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
Then
$c_{P,\,\mu}^{\,\nu}(a)=0$ unless $\mu\subseteq \nu$,
and $c_{P,\,\mu}^{\,\mu}(a)=P(a_{\mu})$. Moreover, if
$\mu\subseteq \nu$, then
\betan
c_{P,\,\mu}^{\,\nu}(a)=\frac{1}{|a_{\nu}|-|a_{\mu}|}
\Bigg(\sum_{\mu^+,\,\mu\to\mu^+}c_{P,\,\mu^+}^{\,\nu}(a)
-\sum_{\nu^-,\,\nu^-\to\nu}c_{P,\,\mu}^{\,\nu^-}(a)\Bigg).
\end{equation*}
The same coefficient can also be found by the
formula
\betaql{rati}
c_{P,\,\mu}^{\,\nu}(a)=\sum_{R}\sum_{k=0}^{l}
\frac{P(a_{\rho^{(k)}})}{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(l)}}|)},
\end{equation}
summed over all sequences of partitions $R$
of the form
\betan
\mu=\rho^{(0)}\to\rho^{(1)}\to
\dots\to\rho^{(l-1)}\to\rho^{(l)}=\nu,
\end{equation*}
where the symbol $\wedge$ indicates that the zero factor
should be skipped.
\qed
\end{prop}

\section{Cauchy identities and dual Schur functions}
\lambdabel{sec:dua}
\setcounter{equation}{0}

\subsection{Definition of dual Schur functions and
Cauchy identities}

We let $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ denote the ring
of formal series of the symmetric functions in
the set of indeterminates $x=(x_1,x_2,\dots)$
with coefficients in $\mathbb{Q}\tss[a]$. More precisely,
\betaql{defla}
\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)=\Big\{\sum_{\lambda\in\mathcal{P}} c_{\lambda}(a)\, s_{\lambda}(x)
\ |\ c_{\lambda}(a)\in \mathbb{Q}\tss[a]\Big\}.
\end{equation}
The Schur functions $s_{\lambda}(x)$ can certainly
be replaced here by any other classical basis of $\Lambda$ parameterized
by the set of partitions $\mathcal{P}$. We will use the symbol
$\widehat\Lambda_n=\widehat\Lambda_n(x{\tss|\hspace{-1.5pt}|\tss} a)$ to indicate
the ring defined as in \eqref{defla}
for the case of the finite set of variables $x=(x_1,\dots,x_n)$.
An element of $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ can be viewed as a sequence
of elements of $\widehat\Lambda_n$ with $n=0,1,\dots$,
consistent with respect to the evaluation maps
\betan
\mathfrak{p}si_n: \widehat\Lambda_n\to \widehat\Lambda_{n-1},\qquad
Q(x_1,\dots,x_n)\mapsto Q(x_1,\dots,x_{n-1},0).
\end{equation*}

For any $n$-tuple of nonnegative integers
$\beta=(\beta_1,\dots,\beta_n)$ set
\betan
A_{\beta}(x, a)=\delta^{}t\big[(x_i,a)^{\beta_j}
\,s(1-a^{}_{n-\beta_j-1}x_i)(1-a^{}_{n-\beta_j-2}\, x_i)\dots
(1-a^{}_{1-\beta_j}x_i)\big]_{i,j=1}^n,
\end{equation*}
where $(x_i,a)^{0}=1$ and
\betaql{dumon}
(x_i,a)^{r}=\frac{x_i^r}
{(1-a^{}_0\,s x_i)(1-a^{}_{-1}\,s x_i)\dots (1-a^{}_{1-r}\,s x_i)},
\qquad r\mathfrak{g}eqslant 1.
\end{equation}

Let $\lambda=(\lambda_1,\dots,\lambda_n)$ be a partition of length
not exceeding $n$. Denote by $d$ the number of boxes
on the diagonal of $\lambda$. That is, $d$ is
determined by
the condition that
$\lambda_{d+1}\leqslant d\leqslant \lambda_d$.
The $(i,j)$ entry $A_{ij}$ of the determinant
$A_{\lambda+\delta^{}}(x, a)$ can be written
more explicitly as
\betan
A_{ij}=\betagin{cases}\dfrac{x_i^{\lambda_j+n-j}}
{(1-a^{}_0\,s x_i)(1-a^{}_{-1}\,s x_i)\dots (1-a^{}_{j-\lambda_j}\,s x_i)}
\quad&\text{for}\quad j=1,\dots,d,\\[1.5em]
x_i^{\lambda_j+n-j}\,
(1-a^{}_1\,s x_i)(1-a^{}_2\,s x_i)\dots (1-a^{}_{j-\lambda_j-1}\,s x_i)
\quad&\text{for}\quad j=d+1,\dots,n.
\end{cases}
\end{equation*}
Observe that the determinant $A_{\delta^{}}(x, a)$
corresponding to the empty partition equals the Vandermonde
determinant,
\betan
A_{\delta^{}}(x, a)=\mathfrak{p}rod_{1\leqslant i<j\leqslant n}(x_i-x_j).
\end{equation*}
Hence, the formula
\betaql{dusf}
\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\frac{A_{\lambda+\delta^{}}(x, a)}{A_{\delta^{}}(x, a)}
\end{equation}
defines an element of the ring $\widehat\Lambda_n$. Furthermore,
setting $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=0$ if the length of $\lambda$
exceeds the number of the $x$ variables, we obtain that
the evaluation of the element $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\widehat\Lambda_n$ at
$x_n=0$ yields the corresponding element of $\widehat\Lambda_{n-1}$
associated with $\lambda$.
Thus, the sequence $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\widehat\Lambda_n$
for $n=0,1,\dots$
defines an element $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ of $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
which we call the {\it dual Schur function\/}.
The lowest degree component of $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
in $x$ coincides with the Schur function $s_{\lambda}(x)$.
Moreover, if $a$ is specialized to
the sequence of zeros, then
$\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ specializes to $s_{\lambda}(x)$.

Now we prove an analogue of the Cauchy identity
involving the double and dual Schur functions.
Consider one more set of variables $y=(y_1,y_2,\dots)$.

\begin{thm}\lambdabel{thm:cch}
The following identity holds
\betaql{cchid}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
\end{thm}

\begin{proof}
We use a modification
of the argument applied in \cite[Section~I.4]{m:sfh}
for the proof of the classical Cauchy identity
(see formula (4.3) there).
As we pointed out in Section~\ref{subsec:dpr},
it will be sufficient to prove the identity in the case
of finite sets of variables $x=(x_1,\dots,x_n)$ and
$y=(y_1,\dots,y_n)$. We have
\betaql{adexy}
A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)\, A_{\delta^{}}(y, a)
\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\mathfrak{g}amma} A_{\mathfrak{g}amma}(x{\tss|\hspace{-1.5pt}|\tss} a)\, A_{\mathfrak{g}amma}(y, a),
\end{equation}
summed over $n$-tuples $\mathfrak{g}amma=(\mathfrak{g}amma_1,\dots,\mathfrak{g}amma_n)$
with $\mathfrak{g}amma_1>\dots>\mathfrak{g}amma_n\mathfrak{g}eqslant 0$. Since
\betan
A_{\mathfrak{g}amma}(y, a)=\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\,s\sigma\,
\mathfrak{p}rod_{i=1}^n (y_i,a)^{\mathfrak{g}amma_{\sigma(i)}}
\,s(1-a^{}_{n-\mathfrak{g}amma_{\sigma(i)}-1}y_i)\dots(1-a^{}_{1-\mathfrak{g}amma_{\sigma(i)}}y_i)
\end{equation*}
and $A_{\mathfrak{g}amma}(x{\tss|\hspace{-1.5pt}|\tss} a)$ is skew-symmetric under
permutations of the components of $\mathfrak{g}amma$, we can write
\eqref{adexy} in the form
\betaql{abexa}
\sum_{\beta} A_{\beta}(x{\tss|\hspace{-1.5pt}|\tss} a)
\mathfrak{p}rod_{i=1}^n (y_i,a)^{\beta_i}
\,s(1-a^{}_{n-\beta_i-1}y_i)\dots(1-a^{}_{1-\beta_i}y_i),
\end{equation}
summed over $n$-tuples $\beta=(\beta_1,\dots,\beta_n)$
on nonnegative integers. Due to the Jacobi--Trudi
formula \eqref{jt}, we have
\betan
A_{\beta}(x{\tss|\hspace{-1.5pt}|\tss} a)=A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\,s\sigma\cdot
h_{\beta_{\sigma(1)}-n+1}(x{\tss|\hspace{-1.5pt}|\tss} a)\dots h_{\beta_{\sigma(n)}}(x{\tss|\hspace{-1.5pt}|\tss} \tau^{n-1}a).
\end{equation*}
Hence, \eqref{abexa} becomes
\betagin{multline}
\lambdabel{axy}
A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)
\sum_{\alpha}h_{\alpha_1}(x{\tss|\hspace{-1.5pt}|\tss} a)\dots h_{\alpha_n}(x{\tss|\hspace{-1.5pt}|\tss} \tau^{n-1}a)\\
{}\times\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\,s\sigma\cdot
\mathfrak{p}rod_{i=1}^n (y_{\sigma(i)},a)^{\alpha_i+n-i}
\,s(1-a^{}_{i-\alpha_i-1}y_{\sigma(i)})
\dots(1-a^{}_{i-\alpha_i-n+1}y_{\sigma(i)}),
\end{multline}
summed over $n$-tuples $\alpha=(\alpha_1,\dots,\alpha_n)$
on nonnegative integers. However, using \eqref{genh},
for each $i=1,\dots,n$ we obtain
\betan
\begin{aligned}
\sum_{k=0}^{\infty}& h_k(x{\tss|\hspace{-1.5pt}|\tss} \tau^{i-1}a)\, (z,a)^{k+n-i}
\,s(1-a^{}_{i-\alpha_i-1}z)
\dots(1-a^{}_{i-\alpha_i-n+1}z)\\
{}&=z^{n-i}\, (1-a_1\,s z)\dots(1-a_{i-1}\,s z)
\sum_{k=0}^{\infty} h_k(x{\tss|\hspace{-1.5pt}|\tss} \tau^{i-1}a)\, (z,\tau^{i-1}a)^k\\
{}&=z^{n-i}\, (1-a_1\,s z)\dots(1-a_{i-1}\,s z)
\mathfrak{p}rod_{r=1}^{n}\frac{1-a_{i+r-1}\,s z}{1-x_r\,s z},
\end{aligned}
\end{equation*}
where we put $z=y_{\sigma(i)}$. Therefore, \eqref{axy} simplifies to
\betan
\begin{aligned}
A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)&\,
\mathfrak{p}rod_{i,j=1}^{n}\frac{1-a_{i}\,s y_j}{1-x_i\,s y_j}
\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\,s\sigma\cdot
\mathfrak{p}rod_{i=1}^n y_{\sigma(i)}^{n-i}
\,(1-a^{}_{n+1}y_{\sigma(i)})
\dots(1-a^{}_{n+i-1}y_{\sigma(i)})\\
{}&=A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)\,s A_{\delta^{}}(y, a)\,
\mathfrak{p}rod_{i,j=1}^{n}\frac{1-a_{i}\,s y_j}{1-x_i\,s y_j},
\end{aligned}
\end{equation*}
thus completing the proof.
\end{proof}

Let $z=(z_1,z_2,\dots)$ be another set of variables.

\begin{cor}\lambdabel{cor:ssduco}
The following identity holds
\betan
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+y_i\, z_j}{1-x_i\, z_j}
=\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(z{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
\end{cor}

\begin{proof}
Observe that
the elements $\widehat s_{\lambda}(z{\tss|\hspace{-1.5pt}|\tss} a)\in \widehat\Lambda(z{\tss|\hspace{-1.5pt}|\tss} a)$ are uniquely
determined by this relation. Hence, the claim follows
by the application of Proposition~\ref{prop:imisom}
and Theorem~\ref{thm:cch}.
\end{proof}

Some other identities of this kind are immediate from
the symmetry property \eqref{symprsu}
and Corollary~\ref{cor:ssduco}.

\begin{cor}\lambdabel{cor:duco}
We have the identities
\betan
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+x_i\, z_j}{1-y_i\, z_j}
=\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda'}(z{\tss|\hspace{-1.5pt}|\tss} a')
\end{equation*}
and
\betan
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+x_i\, y_j}{1+a_i\, y_j}
=\sum_{\lambda\in\mathcal{P}} s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda'}(y{\tss|\hspace{-1.5pt}|\tss} a').
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{cor}

\subsection{Combinatorial presentation}

Given a skew diagram $\theta$, introduce the
corresponding {\it skew dual Schur function\/}
$\widehat s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)$ by the formula
\betaql{skewd}
\widehat s_{\theta}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{T}
\mathfrak{p}rod_{\alpha\in\theta}X_{T(\alpha)}(a^{}_{-c(\alpha)+1},a^{}_{-c(\alpha)}),
\end{equation}
summed over the reverse $\theta$-tableaux $T$, where
\betan
X_i(g,h)=
\frac{x_i\,(1-g\, x_{i-1})\dots (1-g\, x_1)}
{(1-h\, x_{i})\dots (1-h\, x_1)}.
\end{equation*}

\begin{thm}\lambdabel{thm:tab}
For any partition $\mu$ the following identity holds
\betaql{skcid}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_{i-\mu_i}\, y_j}{1-x_i\, y_j}
\, s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\nu} s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation}
summed over partitions $\nu$ containing $\mu$.
In particular, if
$\theta=\lambda$ is a normal {\rm(}nonskew{\rm)} diagram, then
the dual Schur function $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ admits
the tableau presentation
\eqref{skewd}.
\end{thm}

\begin{proof}
It will be sufficient to consider the case where
the set of variables $y$ is finite,
$y=(y_1,\dots,y_n)$. We will argue by induction on $n$
and suppose that $n\mathfrak{g}eqslant 1$. By the induction hypothesis,
the identity \eqref{skcid} holds
for the set of variables $y'=(y_2,\dots,y_n)$.
Hence, we need to verify that
\betan
\mathfrak{p}rod_{i\mathfrak{g}eqslant 1}\frac{1-a_{i-\mu_i}\, y_1}{1-x_i\, y_1}
\,\sum_{\lambda} s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda/\mu}(y'{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\nu} s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
However, due to \eqref{genh},
\betan
\mathfrak{p}rod_{i\mathfrak{g}eqslant 1}\frac{1-a_{i-\mu_i}\, y_1}{1-x_i\, y_1}
=\sum_{k=0}^{\infty} \frac{h_k(x{\tss|\hspace{-1.5pt}|\tss} a^{\mu})\, y_1^k}
{(1-a_0\,s y_1)\dots(1-a_{-k+1}\,s y_1)},
\end{equation*}
where $a^{\mu}$ denotes the sequence of parameters such that
$(a^{\mu})_i=a_{i-\mu_i}$ for $i\mathfrak{g}eqslant 1$ and
$(a^{\mu})_i=a_i$ for $i\leqslant 0$. Now define
polynomials $c_{\lambda,(k)}^{\nu}(a,a^{\mu})\in\mathbb{Q}\tss[a]$ by
the expansion
\betan
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, h_k(x{\tss|\hspace{-1.5pt}|\tss} a^{\mu})=\sum_{\nu}
c_{\lambda,(k)}^{\nu}(a,a^{\mu})\, s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
Hence, the claim will follow if we show that
\betaql{snumu}
\widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda,\, k}c_{\lambda,(k)}^{\nu}(a,a^{\mu})
\, \widehat s_{\lambda/\mu}(y'{\tss|\hspace{-1.5pt}|\tss} a)\, \frac{y_1^k}
{(1-a_0\,s y_1)\dots(1-a_{-k+1}\,s y_1)}.
\end{equation}
The definition \eqref{skewd} of the skew dual Schur
functions implies that
\betan
\widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}
\widehat s_{\lambda/\mu}(y'{\tss|\hspace{-1.5pt}|\tss} a)\,\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\frac{1-a_{-c(\alpha)+1}\,s y_1}{1-a_{-c(\alpha)}\,s y_1}
\,\mathfrak{p}rod_{\beta\in\nu/\lambda}
\frac{y_1}{1-a_{-c(\alpha)}\,s y_1},
\end{equation*}
summed over diagrams $\lambda$ such that
$\mu\subseteq\lambda\subseteq\nu$ and $\nu/\lambda$ is a horizontal
strip (i.e., every column of this diagram contains
at most one box). Therefore, \eqref{snumu}
will follow from the relation
\betan
\begin{aligned}
\sum_{k}&c_{\lambda,(k)}^{\nu}(a,a^{\mu})
\,\frac{y_1^k}
{(1-a_0\,s y_1)\dots(1-a_{-k+1}\,s y_1)}\\
{}&=\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\frac{1-a_{-c(\alpha)+1}\,s y_1}{1-a_{-c(\alpha)}\,s y_1}
\,\mathfrak{p}rod_{\beta\in\nu/\lambda}
\frac{y_1}{1-a_{-c(\alpha)}\,s y_1}
\end{aligned}
\end{equation*}
which takes more convenient form after the
substitution $t=y_1^{-1}$:
\betaql{sumts}
\sum_{k}
\,\frac{c_{\lambda,(k)}^{\nu}(a,a^{\mu})}
{(t-a_0)\dots(t-a_{-k+1})}=\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\big(t-a_{-c(\alpha)+1}\big)
\,\mathfrak{p}rod_{\beta\in\nu/\mu}
\big(t-a_{-c(\alpha)}\big)^{-1}.
\end{equation}
We will verify the latter by induction on $|\nu|-|\lambda|$.
Suppose first that $\nu=\lambda$. Then
$c_{\lambda,(k)}^{\lambda}(a,a^{\mu})=h_k(a_{\lambda}{\tss|\hspace{-1.5pt}|\tss} a^{\mu})$
by Proposition~\ref{prop:interp}, and relation \eqref{genh}
implies that
\betan
\sum_{k}
\,\frac{h_k(a_{\lambda}{\tss|\hspace{-1.5pt}|\tss} a^{\mu})}
{(t-a_0)\dots(t-a_{-k+1})}=\mathfrak{p}rod_{i\mathfrak{g}eqslant 1}
\frac{t-a_{i-\mu_i}}{t-a_{i-\lambda_i}}.
\end{equation*}
This expression coincides with
\betan
\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\frac{t-a_{-c(\alpha)+1}}{t-a_{-c(\alpha)}},
\end{equation*}
thus verifying \eqref{sumts} in the case under
consideration. Suppose now that $|\nu|-|\lambda|\mathfrak{g}eqslant 1$.
By Proposition~\ref{prop:interp},
we have
\betan
c_{\lambda,(k)}^{\nu}(a,a^{\mu})=\frac{1}{|a_{\nu}|-|a_{\lambda}|}\,
\Bigg(\sum_{\lambda^+,\,\lambda\to\lambda^+}c_{\lambda^+,(k)}^{\nu}(a,a^{\mu})
-\sum_{\nu^-,\,\nu^-\to\nu}c_{\lambda,(k)}^{\nu^-}(a,a^{\mu})\Bigg).
\end{equation*}
Hence, applying the induction hypothesis, we can
write the left hand side of \eqref{sumts} in the form
\betan
\begin{aligned}
\frac{1}{|a_{\nu}|-|a_{\lambda}|}\,
\Big(&\sum_{\lambda^+}\mathfrak{p}rod_{\alpha\in\lambda^+/\mu}
\big(t-a_{-c(\alpha)+1}\big)
\,\mathfrak{p}rod_{\beta\in\nu/\mu}
\big(t-a_{-c(\alpha)}\big)^{-1}\\
{}-{}&\sum_{\nu^-}\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\big(t-a_{-c(\alpha)+1}\big)
\,\mathfrak{p}rod_{\beta\in\nu^-/\mu}
\big(t-a_{-c(\alpha)}\big)^{-1}\Big).
\end{aligned}
\end{equation*}
Since $\nu/\lambda$ is a horizontal strip, we have
\betan
\sum_{\alpha=\lambda^+/\lambda}\big(t-a_{-c(\alpha)+1}\big)-
\sum_{\alpha=\nu/\nu^-}\big(t-a_{-c(\alpha)}\big)=|a_{\nu}|-|a_{\lambda}|,
\end{equation*}
so that the previous expression simplifies to
\betan
\mathfrak{p}rod_{\alpha\in\lambda/\mu}
\big(t-a_{-c(\alpha)+1}\big)
\,\mathfrak{p}rod_{\beta\in\nu/\mu}
\big(t-a_{-c(\alpha)}\big)^{-1}
\end{equation*}
completing the proof of \eqref{sumts}.

The second part of the proposition follows from
Theorem~\ref{thm:cch} and the fact that the
elements $\widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\in\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$ are uniquely
determined by the relation \eqref{cchid}.
\end{proof}

\begin{remark}\lambdabel{rem:aze}
Under the specialization $a_i=0$ the identity
of Theorem~\ref{thm:tab} turns into a particular case
of the identity in \cite[Example~I.5.26]{m:sfh}.
\qed
\end{remark}

Since the skew dual Schur functions are uniquely determined
by the expansion \eqref{skcid}, the following corollary
is immediate from Theorem~\ref{thm:tab}.

\begin{cor}\lambdabel{cor:symme}
The skew dual Schur functions defined in \eqref{skewd}
belong to the ring $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$. In particular,
they are symmetric in the variables $x$.
\qed
\end{cor}

Recall the Littlewood--Richardson polynomials
defined by \eqref{lrpoldef}.

\begin{prop}\lambdabel{prop:skelrp}
For any skew diagram $\nu/\mu$ we have the expansion
\betan
\widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda} c_{\lambda\mu}^{\,s\nu}(a)\,
\widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
\end{prop}

\begin{proof}
We use an argument similar to the one used
in \cite[Section~I.5]{m:sfh}.
Consider the set of variables $(y,y^{\,s\mathfrak{p}rime})$,
where $y=(y_1,y_2,\dots)$
and $y^{\,s\mathfrak{p}rime}=(y^{\,s\mathfrak{p}rime}_1,y^{\,s\mathfrak{p}rime}_2,\dots)$
and assume they are ordered in the way
that each $y_i$ precedes each $y^{\,s\mathfrak{p}rime}_j$. By the tableau
presentation \eqref{skewd} of the dual Schur functions,
we get
\betaql{snuyza}
\widehat s_{\nu}(y,y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu\subseteq\nu}
\widehat s_{\nu/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
On the other hand, by Theorem~\ref{thm:cch},
\betan
\begin{aligned}
{}&\sum_{\nu}s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\nu}(y,y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)=
\mathfrak{p}rod_{i,j\mathfrak{g}eqslant 1}\frac{1-a_i\,s y_j}{1-x_i\,s y_j}
\,\mathfrak{p}rod_{i,k\mathfrak{g}eqslant 1}
\frac{1-a_i\,s y^{\,s\mathfrak{p}rime}_k}{1-x_i\,s y^{\,s\mathfrak{p}rime}_k}\\
{}&=\sum_{\lambda,\,\mu}s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\,
s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\lambda,\,\mu,\,\nu}c_{\lambda\mu}^{\,s\nu}(a)\,
s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\,
\widehat s_{\mu}(y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
\end{aligned}
\end{equation*}
which proves that
\betaql{snulrp}
\widehat s_{\nu}(y,y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\lambda,\,\mu}c_{\lambda\mu}^{\,s\nu}(a)\,
\widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
The desired relation now follows by comparing \eqref{snuyza}
and \eqref{snulrp}.
\end{proof}

\subsection{Jacobi--Trudi-type formulas}

Introduce the {\it dual elementary\/} and {\it complete
symmetric functions\/} by
\betan
\widehat e_k(x{\tss|\hspace{-1.5pt}|\tss} a)=\widehat s_{(1^k)}(x{\tss|\hspace{-1.5pt}|\tss} a),\qquad
\widehat h_k(x{\tss|\hspace{-1.5pt}|\tss} a)=\widehat s_{(k)}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
By Theorem~\ref{thm:tab},
\betan
\begin{aligned}
\widehat e_k(x{\tss|\hspace{-1.5pt}|\tss} a)&=\sum_{i_1>\dots>i_k}
X_{i_1}(a_1,a_0)X_{i_2}(a_2,a_1)\dots
X_{i_k}(a_k,a_{k-1}),\\
\widehat h_k(x{\tss|\hspace{-1.5pt}|\tss} a)&=\sum_{i_1\mathfrak{g}eqslant\dots\mathfrak{g}eqslant i_k}
X_{i_1}(a_1,a_0)X_{i_2}(a_0,a_{-1})\dots
X_{i_k}(a_{-k+2},a_{-k+1}).
\end{aligned}
\end{equation*}

\begin{prop}\lambdabel{prop:gendu}
We have the following generating series formulas
\betan
\begin{aligned}
1+\sum_{k=1}^{\infty}\widehat e_k(x{\tss|\hspace{-1.5pt}|\tss} a)\,
(t+a_0)(t+a_1)\dots (t+a_{k-1})=
\mathfrak{p}rod_{i=1}^{\infty}\frac{1+t\,s x_i}{1-a_0\,s x_i},\\
1+\sum_{k=1}^{\infty}\widehat h_k(x{\tss|\hspace{-1.5pt}|\tss} a)\,
(t-a_1)(t-a_0)\dots (t-a_{-k+2})=
\mathfrak{p}rod_{i=1}^{\infty}\frac{1-a_1\,s x_i}{1-t\,s x_i}.
\end{aligned}
\end{equation*}
\end{prop}

\begin{proof}
The first relation follows from
the second identity in Corollary~\ref{cor:duco}
by taking $x=(t)$ and then replacing $a$ by $a'$ and
$y_i$ by $x_i$ for all $i$. Similarly,
the second relation follows from Theorem~\ref{thm:cch}
by taking $x=(t)$ and replacing
$y_i$ by $x_i$.
\end{proof}

We can now prove an analogue of the Jacobi--Trudi formula
for the dual Schur functions.

\begin{prop}\lambdabel{prop:jtdu}
If $\lambda$ and $\mu$ are
partitions of length not exceeding $n$, then
\betaql{jtdu}
\widehat s_{\lambda/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\delta^{}t\big[\,s\widehat h_{\lambda_i-\mu_j-i+j}
(x{\tss|\hspace{-1.5pt}|\tss} \tau^{\,s -\mu_j+j-1}\,s a)\big]_{i,j=1}^n.
\end{equation}
\end{prop}

\begin{proof}
Apply Theorem~\ref{thm:tab} for the finite set
of variables $x=(x_1,\dots,x_n)$ and multiply both
sides of \eqref{skcid} by $A_{\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)$.
This gives
\betaql{pramu}
\mathfrak{p}rod_{j\mathfrak{g}eqslant 1}\mathfrak{p}rod_{i=1}^n
\frac{1-a_{i-\mu_i}\, y_j}{1-x_i\, y_j}
\, A_{\mu+\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\lambda} A_{\lambda+\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\lambda/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
For any $\sigma\in\mathfrak S_n$ we have
\betan
\mathfrak{p}rod_{j\mathfrak{g}eqslant 1}\mathfrak{p}rod_{i=1}^n
\frac{1-a_{i-\mu_i}\, y_j}{1-x_i\, y_j}
=\mathfrak{p}rod_{j\mathfrak{g}eqslant 1}\mathfrak{p}rod_{i=1}^n
\frac{1-a_{i-\mu_i}\, y_j}{1-x_{\sigma(i)}\, y_j}.
\end{equation*}
By the second formula of Proposition~\ref{prop:gendu},
\betan
\mathfrak{p}rod_{j\mathfrak{g}eqslant 1}
\frac{1-a_{i-\mu_i}\, y_j}{1-x_{\sigma(i)}\, y_j}
=\sum_{k=0}^{\infty}
\widehat h_k(y{\tss|\hspace{-1.5pt}|\tss}\tau^{-\mu_i+i-1}a)\,
(x_{\sigma(i)}-a_{i-\mu_i})\dots(x_{\sigma(i)}-a_{i-\mu_i-k+1}).
\end{equation*}
Since
\betan
A_{\mu+\delta^{}}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\sigma\cdot
(x_{\sigma(1)}{\tss|\hspace{-1.5pt}|\tss} a)^{\mu_1+n-1}\dots (x_{\sigma(n)}{\tss|\hspace{-1.5pt}|\tss} a)^{\mu_n} ,
\end{equation*}
the left hand side of \eqref{pramu}
can be written in the form
\betan
\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\sigma\,
\mathfrak{p}rod_{i=1}^n\sum_{k_i=0}^{\infty}
(x_{\sigma(i)}-a_n)\dots(x_{\sigma(i)}-a_{i-\mu_i-k_i+1})
\, h_{k_i}(y,\tau^{-\mu_i+i-1}a).
\end{equation*}
Hence, comparing the coefficients
of $(x_{1}{\tss|\hspace{-1.5pt}|\tss} a)^{\lambda_1+n-1}\dots (x_n{\tss|\hspace{-1.5pt}|\tss} a)^{\lambda_n}$
on both sides of \eqref{pramu}, we get
\betan
\widehat s_{\lambda/\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\rho\in\mathfrak S_n}
 {\rm sgn}\ts\rho\,\mathfrak{p}rod_{i=1}^n \widehat h_{\lambda_i-\mu_{\rho(i)}-i+\rho(i)}
(y{\tss|\hspace{-1.5pt}|\tss}\tau^{-\mu_{\rho(i)}+\rho(i)-1}a),
\end{equation*}
as required.
\end{proof}

Proposition~\ref{prop:jtdu} implies that the
dual Schur functions may be regarded as a specialization
of the generalized Schur functions described in
\cite[9th~Variation]{m:sf}. Namely,
in the notation of that paper, specialize the variables
$h_{rs}$ by
\betaql{hrsspe}
h_{rs}=\widehat h_r(x{\tss|\hspace{-1.5pt}|\tss}\tau^{-s}a),\qquad r\mathfrak{g}eqslant 1,\quad s\in\mathbb{Z}\tss.
\end{equation}
Then the Schur functions $s_{\lambda/\mu}$ of \cite{m:sf}
become $\widehat s_{\lambda/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$.
Hence the following corollaries are immediate
from $(9.6')$ and $(9.7)$ in \cite{m:sf}
and Proposition~\ref{prop:jtdu}.
The first of them is an analogue of the
N\"{a}gelsbach--Kostka formula.

\begin{cor}\lambdabel{cor:nkdu}
If $\lambda$ and $\mu$ are
partitions such that the lengths
of $\lambda'$ and $\mu'$ do not exceed $m$, then
\betaql{nkdu}
\widehat s_{\lambda/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\delta^{}t\big[\widehat e_{\lambda'_i-\mu'_j-i+j}
(x{\tss|\hspace{-1.5pt}|\tss}\tau^{\,s \mu'_j-j+1}\,s a)\big]_{i,j=1}^m.
\end{equation}
\end{cor}

Suppose that $\lambda$ is a diagram with $d$ boxes
on the main diagonal. Write $\lambda$ in the
Frobenius notation
\betan
\lambda=(\alpha_1,\dots,\alpha_d\,s|\,s\beta_1,\dots,\beta_d)=(\alpha\,s|\,s\beta),
\end{equation*}
where $\alpha_i=\lambda_i-i$ and $\beta_i=\lambda'_i-i$.
The following is an analogue of the Giambelli formula.

\begin{cor}\lambdabel{cor:gidu}
We have the identity
\betaql{gidu}
\widehat s_{(\alpha\,s|\,s\beta)}(x{\tss|\hspace{-1.5pt}|\tss} a)
=\delta^{}t\big[\widehat s_{(\alpha_i\,s|\,s\beta_j)}
(x{\tss|\hspace{-1.5pt}|\tss} a)\big]_{i,j=1}^d.
\end{equation}
\end{cor}

\subsection{Expansions in terms of Schur functions}

We will now deduce
expansions of the dual Schur functions in terms
of the Schur functions $s_{\lambda}(x)$
whose coefficients are elements of $\mathbb{Q}\tss[a]$
written explicitly as certain determinants.
In Theorem~\ref{thm:tabre} below we will give
alternative tableau presentations for these coefficients.
Suppose that $\mu$ is a diagram containing $d$ boxes
on the main diagonal.

\begin{prop}\lambdabel{prop:dsfexp}
The dual Schur function $\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ can be written
as the series
\betan
\begin{aligned}
\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}(-1)^{n(\lambda/\mu)}\,
&\delta^{}t\big[h_{\lambda_i-\mu_j-i+j}
(a^{}_0,a^{}_{-1},\dots,a^{}_{j-\mu_j})\big]_{i,j=1}^d\\
{}\times{}&
\delta^{}t\big[e_{\lambda_i-\mu_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})\big]_{i,j=d+1}^n
\, s_{\lambda}(x),
\end{aligned}
\end{equation*}
summed over diagrams $\lambda$ which
contain $\mu$ and such that $\lambda$ has
$d$ boxes on the main diagonal, where
$n(\lambda/\mu)$ denotes the total number of boxes in
the diagram $\lambda/\mu$ in rows $d+1,d+2,\dots,n=\ell(\lambda)$.
\end{prop}

\begin{proof}
It will be sufficient to prove the formula for the case
of finite set of variables $x=(x_1,\dots,x_n)$.
We use the definition \eqref{dusf} of the dual Schur
functions. The entries $A_{ij}$ of
the determinant $A_{\mu+\delta^{}}(x, a)$
can be written as
\betan
A_{ij}=\betagin{cases}{\displaystyle\sum_{p_j\mathfrak{g}eqslant 0}
h_{p_j}(a^{}_0,a^{}_{-1},\dots,a^{}_{j-\mu_j})\,
x_i^{\mu_j+p_j+n-j}}
\quad&\text{for}\quad j=1,\dots,d,\\[2em]
{\displaystyle\sum_{p_j\mathfrak{g}eqslant 0}
(-1)^{p_j}\,
e_{p_j}(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})\,
x_i^{\mu_j+p_j+n-j}}
\quad&\text{for}\quad j=d+1,\dots,n.
\end{cases}
\end{equation*}
Hence, \eqref{dusf} gives
\betan
\begin{aligned}
\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)={}&\sum_{p_1,\dots,\, p_n}
\mathfrak{p}rod_{j=1}^d
h_{p_j}(a^{}_0,a^{}_{-1},\dots,a^{}_{j-\mu_j})
\mathfrak{p}rod_{j=d+1}^n
(-1)^{p_j}\, e_{p_j}(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})\\[1em]
{}&{}\times\delta^{}t[x_i^{\mu_j+p_j+n-j}]/\delta^{}t[x_i^{n-j}].
\end{aligned}
\end{equation*}
The ratio of the determinants in this formula
is nonzero only if
\betan
\mu_{\sigma(j)}+p_{\sigma(j)}+n-{\sigma(j)}=\lambda_j+n-j,\qquad j=1,\dots,n,
\end{equation*}
for some diagram $\lambda$ containing $\mu$ and some
permutation $\sigma$ of the set $\{1,\dots,n\}$.
Moreover, since $e_{p_j}(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})=0$
for $p_j>j-\mu_j-1$, the number of diagonal boxes
in $\lambda$ equals $d$.
The ratio can then be written as
\betan
\delta^{}t[x_i^{\mu_j+p_j+n-j}]/\delta^{}t[x_i^{n-j}]= {\rm sgn}\ts\sigma\cdot
s_{\lambda}(x),
\end{equation*}
which gives the desired formula for the coefficients.
\end{proof}

\begin{cor}\lambdabel{cor:sone}
Using the Frobenius notation $(\alpha\,s|\,s\beta)$
for the hook diagram $(\alpha+1,1^{\beta})$, we
have
\betan
\widehat s_{(\alpha\,s|\,s\beta)}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{p,\, q\mathfrak{g}eqslant 0}(-1)^q\,
h_p(a^{}_0,a^{}_{-1},\dots,a^{}_{-\alpha})\,
h_q(a^{}_1,a^{}_2,\dots,a^{}_{\beta+1})\,
s^{}_{(\alpha+p\,s|\,s\beta+q)}(x).
\end{equation*}
\end{cor}

\begin{proof}
By Proposition~\ref{prop:dsfexp}, the coefficient of
$s^{}_{(\alpha+p\,s|\,s\beta+q)}(x)$ in the expansion
of the dual Schur function $\widehat s_{(\alpha\,s|\,s\beta)}(x{\tss|\hspace{-1.5pt}|\tss} a)$ equals
\betaql{hdete}
(-1)^q\,
h_p(a^{}_0,a^{}_{-1},\dots,a^{}_{-\alpha})\,
\delta^{}t\big[e_{j-i+1}
(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j})\big]_{i,j=1}^q.
\end{equation}
Using the relations for the elementary symmetric
polynomials
\betan
e_k(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j})
=e_k(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j-1})+
e_{k-1}(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j-1})\, a^{}_{\beta+j},
\end{equation*}
it is not difficult to bring the determinant
which occurs in \eqref{hdete} to the form
\betaql{deteq}
\delta^{}t\big[e_{j-i+1}
(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j})\big]_{i,j=1}^q
=\delta^{}t\big[e_{j-i+1}
(a^{}_1,a^{}_2,\dots,a^{}_{\beta+1})\big]_{i,j=1}^q.
\end{equation}
Indeed, denote by $C_1,\dots,C_q$ the columns of the
$q\times q$ matrix which occurs
on the left hand side. Now replace
$C_j$ by $C_j-a^{}_{\beta+j}\, C_{j-1}$
consequently for $j=q,q-1,\dots,2$. These operations leave
the determinant of the matrix unchanged, while for $j\mathfrak{g}eqslant 2$ the
$(i,j)$ entry of the new matrix equals
$e_{j-i+1}
(a^{}_1,a^{}_2,\dots,a^{}_{\beta+j-1})$. Applying similar
column operations to the new matrix and using obvious induction
we will bring its determinant
to the form which occurs on the right hand side of \eqref{deteq}.
However, this determinant
coincides with $h_q(a^{}_1,a^{}_2,\dots,a^{}_{\beta+1})$
due to the N\"{a}gelsbach--Kostka formula (i.e., \eqref{nkdu}
with the zero sequence $a$; that is, $a_i=0$ for all $i\in\mathbb{Z}\tss$).
\end{proof}

\betax\lambdabel{ex:sone}
The dual Schur function corresponding
to the single box diagram is given by
\betan
\widehat s_{(1)}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{p,\, q\mathfrak{g}eqslant 0}
(-1)^q\, a_0^p\, a_1^q\,
s^{}_{(p\,s|\,s q)}(x).
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{example}

Recall that the involution
$\omega^{}:\Lambda\to\Lambda$ on the ring of symmetric functions
in $x$ takes $s_{\lambda}(x)$ to $s_{\lambda'}(x)$; see
\cite[Section~I.2]{m:sfh} or Section~\ref{sec:def} above.
Let us extend $\omega^{}$ to the $\mathbb{Q}\tss[a]$-linear
involution
\betaql{whome}
\widehat\omega^{}:\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to \widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a),\qquad
\sum_{\lambda\in\mathcal{P}} c_{\lambda}(a)\, s_{\lambda}(x)\mapsto
\sum_{\lambda\in\mathcal{P}} c_{\lambda}(a)\, s_{\lambda'}(x),
\end{equation}
where $c_{\lambda}(a)\in\mathbb{Q}\tss[a]$.
We will find the images
of the dual Schur functions under $\widehat\omega^{}$.
As before, by $a'$ we denote the sequence of variables
such that $(a')_i=-a_{-i+1}$ for all $i\in\mathbb{Z}\tss$.

\begin{cor}\lambdabel{cor:omhat}
For any skew diagram $\lambda/\mu$ we have
\betaql{imomdu}
\widehat\omega^{}:\widehat s_{\lambda/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto  \widehat s_{\lambda'/\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a').
\end{equation}
\end{cor}

\begin{proof}
By Corollary~\ref{cor:sone}, for any $m\in\mathbb{Z}\tss$
\betan
\widehat\omega^{}:
\widehat s_{(\alpha\,s|\,s\beta)}(x{\tss|\hspace{-1.5pt}|\tss}\tau^ma)\mapsto
\widehat s_{(\beta\,s|\,s\alpha)}(x{\tss|\hspace{-1.5pt}|\tss}\tau^{-m}a').
\end{equation*}
In particular,
\betan
\widehat\omega^{}:
\widehat h_k(x{\tss|\hspace{-1.5pt}|\tss}\tau^m a)\mapsto \widehat e_k(x{\tss|\hspace{-1.5pt}|\tss}\tau^{-m}a'),
\qquad k\mathfrak{g}eqslant 0.
\end{equation*}
The statement now follows from \eqref{jtdu} and \eqref{nkdu}.
\end{proof}

Note that \eqref{imomdu} with $\mu=\varnothing$ also
follows from Corollary~\ref{cor:sone}
and the Giambelli formula \eqref{gidu}.

\begin{remark}\lambdabel{rem:athom}
The involution $\widehat\omega^{}$ does not coincide
with the involution introduced in \cite[(9.6)]{m:sfh}.
The latter is defined on the ring generated by the
elements $h_{rs}$ and
takes the generalized Schur function
$s_{\lambda/\mu}$ to $s_{\lambda'/\mu'}$. Therefore, under the
specialization \eqref{hrsspe}, the image of
$\widehat s_{\lambda/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ would be $\widehat s_{\lambda'/\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a)$
which is different from \eqref{imomdu}.
\qed
\end{remark}

We can now derive an alternative expansion
of the dual Schur functions in terms of the Schur functions
$s_{\lambda}(x)$; cf. Proposition~\ref{prop:dsfexp}.
Suppose that $\lambda$
is a diagram
which contains $\mu$ and such that $\mu$ and $\lambda$ have the
same number of boxes $d$ on the diagonal.
By a {\it hook $\lambda/\mu$-tableau\/}
$T$ we will mean a tableau obtained by filling in the boxes
of $\lambda/\mu$ with integers in the following way.
The entries in the first $d$ rows
weakly increase along the rows and strictly increase
down the columns, and all entries in row $i$
belong to the set $\{i-\mu_i,\dots,-1,0\}$ for
$i=1,\dots,d$;
the entries in the first $d$ columns
weakly decrease down the columns and strictly decrease
along the rows, and all entries in column $j$
belong to the set $\{1,2,\dots,\mu'_j-j+1\}$ for
$j=1,\dots,d$. Then we define the corresponding flagged
Schur function $\varphi_{\lambda/\mu}(a)$ by the formula
\betan
\varphi_{\lambda/\mu}(a)=\sum_T \mathfrak{p}rod_{\alpha\in\lambda/\mu} a^{}_{T(\alpha)},
\end{equation*}
summed over the hook $\lambda/\mu$-tableaux $T$.

\begin{thm}\lambdabel{thm:tabre}
Let $\mu$ be a diagram and let $d$ be the number
of boxes on the main diagonal of $\mu$.
We have the expansion of the dual Schur function
$\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$
\betan
\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}(-1)^{n(\lambda/\mu)}\,
\varphi_{\lambda/\mu}(a)\, s_{\lambda}(x),
\end{equation*}
summed over diagrams $\lambda$ which
contain $\mu$ and such that $\lambda$ has
$d$ boxes on the main diagonal, where
$n(\lambda/\mu)$ denotes the total number of boxes in
the diagram $\lambda/\mu$ in rows $d+1,d+2,\dots$.
\end{thm}

\begin{proof}
Consider the expansions of $\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and
$\widehat s_{\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a')$
provided by Proposition~\ref{prop:dsfexp}.
By Corollary~\ref{cor:omhat}, $\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ is the image
of $\widehat s_{\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a')$ under the involution $\widehat\omega^{}$.
Since $\widehat\omega^{}:s_{\lambda}(x)\mapsto s_{\lambda'}(x)$,
taking $\lambda_i=\mu_i$ for $i=1,\dots,d$ and comparing
the coefficients of $s_{\lambda}(x)$ in
the expansions of $\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and
$\widehat\omega^{}\big(\widehat s_{\mu'}(x{\tss|\hspace{-1.5pt}|\tss} a')\big)$,
we can conclude that
\betagin{multline}
(-1)^{n(\lambda/\mu)}
\delta^{}t\big[e_{\lambda_i-\mu_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})\big]_{i,j\mathfrak{g}eqslant d+1}\\
=\delta^{}t\big[h_{\lambda'_i-\mu'_j-i+j}
(a'_0,a'_{-1},\dots,a'_{j-\mu'_j})\big]_{i,j=1}^d
\nonumber
\end{multline}
so that
\betan
\delta^{}t\big[e_{\lambda_i-\mu_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{j-\mu_j-1})\big]_{i,j\mathfrak{g}eqslant d+1}
=\delta^{}t\big[h_{\lambda'_i-\mu'_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{\mu'_j-j+1})\big]_{i,j=1}^d.
\end{equation*}
On the other hand, if $\lambda$ is a diagram containing
$\mu$ and such that $\lambda$ has
$d$ boxes on the main diagonal,
both determinants
\betan
\delta^{}t\big[h_{\lambda_i-\mu_j-i+j}
(a^{}_0,a^{}_{-1},\dots,a^{}_{j-\mu_j})\big]_{i,j=1}^d,\quad
\delta^{}t\big[h_{\lambda'_i-\mu'_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{\mu'_j-j+1})\big]_{i,j=1}^d
\end{equation*}
coincide with
the respective `row-flagged Schur functions'
of \cite[(8.2)]{m:sf},
\cite{w:fs}, and they admit
the required tableau presentations.
\end{proof}

It is clear from the definition of the
flagged Schur function $\varphi_{\lambda/\mu}(a)$ that it
can be written as the product of two polynomials.
More precisely, suppose that
the diagram $\lambda$ contains
$\mu$ and both $\lambda$ and $\mu$ have
$d$ boxes on their main diagonals.
Let $(\lambda/\mu)_+$ denote the part of the skew diagram
$\lambda/\mu$ contained in the top $d$ rows.
With this notation,
the hook flagged Schur function $\varphi_{\lambda/\mu}(a)$
can be written as
\betaql{whsufa}
\varphi_{\lambda/\mu}(a)=(-1)^{n(\lambda/\mu)}\,
\varphi_{(\lambda/\mu)_+}(a)\,\varphi_{(\lambda'/\mu')_+}(a').
\end{equation}
In addition to the tableau presentation of
the polynomial $\varphi_{(\lambda/\mu)_+}(a)$ given above,
we can get an alternative presentation based on the
column-flagged Schur functions; see \cite[$(8.2')$]{m:sf},
\cite{w:fs}. Due to \eqref{whsufa}, this also gives
alternative formulas for the
coefficients in the expansion of $\widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$.

\begin{cor}\lambdabel{cor:anexpa}
We have the tableau presentation
\betan
\varphi_{(\lambda/\mu)_+}(a)=
\sum_T \mathfrak{p}rod_{\alpha\in(\lambda/\mu)_+} a^{}_{T(\alpha)},
\end{equation*}
summed over the $(\lambda/\mu)_+$-tableaux $T$
whose entries in column $j$ belong to the set
$\{0,-1,\dots,-j+\mu'_j+2\}$ for $j\mathfrak{g}eqslant d+1$,
and the entries weakly increase along
the rows and strictly increase down the columns.
\qed
\end{cor}

Our next goal is to derive the inverse formulas
expressing the Schur functions $s_{\mu}(x)$
as a series of the dual Schur functions $\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$.

\begin{prop}\lambdabel{prop:inve}
We have the expansion
\betan
\begin{aligned}
s_{\mu}(x)=\sum_{\lambda}(-1)^{m(\lambda/\mu)}\,
&\delta^{}t\big[e_{\lambda_i-\mu_j-i+j}
(a^{}_0,a^{}_{-1},\dots,a^{}_{i-\lambda_i+1})\big]_{i,j=1}^d\\
{}\times{}&
\delta^{}t\big[h_{\lambda_i-\mu_j-i+j}
(a^{}_1,a^{}_2,\dots,a^{}_{i-\lambda_i})\big]_{i,j\mathfrak{g}eqslant d+1}
\, \widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{aligned}
\end{equation*}
summed over diagrams $\lambda$ which
contain $\mu$ and such that $\lambda$ has
$d$ boxes on the main diagonal,
where $m(\lambda/\mu)$ denotes the total number of boxes in
the diagram $\lambda/\mu$ in rows $1,\dots,d$.
\end{prop}

\begin{proof}
We will work with a finite set of
variables $x=(x_1,\dots,x_n)$.
The one variable specialization of \eqref{genh} gives
\betan
1+\sum_{k=1}^{\infty} \frac{(x-a_1)(x-a_0)\dots (x-a_{-k+2})\, t^k}
{(1-a_0\,s t)\dots(1-a_{-k+1}\,s t)}=
\frac{1-a_1\,s t}{1-x\,s t}.
\end{equation*}
This implies
\betaql{decxt}
\sum_{k=1}^{\infty} \frac{(x-a_0)\dots (x-a_{-k+2})\, t^k}
{(1-a_0\,s t)\dots(1-a_{-k+1}\,s t)}=
\frac{t}{1-x\,s t}.
\end{equation}
Writing
\betan
(x-a_0)\dots (x-a_{-k+2})=\sum_{i=1}^k(-1)^{k-i}
e_{k-i}(a_0,a_{-1},\dots,a_{-k+2})\, x^{i-1}
\end{equation*}
and comparing the coefficients of $x^{r-1}$ on both sides
of \eqref{decxt} we come to the relation
\betaql{tpor}
t^{\,s r}=\sum_{k=r}^{\infty}
\frac{(-1)^{k-r}\, e_{k-r}(a_0,a_{-1},\dots,a_{-k+2})\, t^k}
{(1-a_0\,s t)\dots(1-a_{-k+1}\,s t)},\qquad r\mathfrak{g}eqslant 1.
\end{equation}
Similarly, writing
\betan
\frac{1}
{(1-a_0\,s t)\dots(1-a_{-k+1}\,s t)}
=\sum_{j=0}^{\infty}
h_{j}(a_0,a_{-1},\dots,a_{-k+1})\, t^{j}
\end{equation*}
and comparing the coefficients of $t^{\,s r+1}$ on both sides
of \eqref{decxt} we come to
\betaql{xpor}
x^r=\sum_{k=0}^r h_{r-k}(a_0,a_{-1},\dots,a_{-k})
\,s(x-a_0)(x-a_{-1})\dots(x-a_{-k+1}),\qquad r\mathfrak{g}eqslant 0.
\end{equation}
Assuming that the length of $\mu$ does not exceed $n$,
represent $s_{\mu}(x)$ as the ratio of determinants
\betan
s_{\mu}(x)=\frac{A_{\mu+\delta^{}}(x)}{A_{\delta^{}}(x)},
\end{equation*}
where
\betan
A_{\alpha}(x)=\delta^{}t\big[x_i^{\alpha_j}\big]_{i,j=1}^n,
\qquad \alpha=(\alpha_1,\dots,\alpha_n).
\end{equation*}
By \eqref{tpor},
for any $j=1,\dots,d$ we have
\betan
x_i^{\mu_j-j+1}=\sum_{p=\mu_j-j+1}^{\infty}
\frac{(-1)^{p-\mu_j+j-1}\, e_{p-\mu_j+j-1}
(a_0,a_{-1},\dots,a_{-p+2})\, x_i^p}
{(1-a_0\,s x_i)\dots(1-a_{-p+1}\,s x_i)}.
\end{equation*}
Similarly, for $j=d+1,\dots,n$ we find from \eqref{xpor}
applied for $x=x_i^{-1}$ and $r=j-\mu_j-1$ that
\betan
x_i^{\mu_j-j+1}=\sum_{p=0}^{j-\mu_j-1}
h_{j-\mu_j-p-1}(a_1,a_2,\dots,a_{p+1})\, x_i^{-p}
\,s(1-a_1 x_i)(1-a_2 x_i)\dots(1-a_p x_i).
\end{equation*}
Multiplying both sides of these relations by $x_i^{n-1}$
we get the respective expansions of $x_i^{\mu_j+n-j}$
which allow us to write
\betan
\begin{aligned}
A_{\mu+\delta^{}}(x)=\sum_{\beta_1,\dots,\, \beta_n}
{}&{}\mathfrak{p}rod_{j=1}^d
(-1)^{\beta_j-\mu_j+j-1}\,
e_{\beta_j-\mu_j+j-1}(a^{}_0,a^{}_{-1},\dots,a^{}_{-\beta_j+2})\\
{}\times{}&{}\mathfrak{p}rod_{j=d+1}^n
\, h_{j-\mu_j-\beta_j-1}(a^{}_1,a^{}_2,\dots,a^{}_{\beta_j+1})
\, A_{\beta}(x, a).
\end{aligned}
\end{equation*}
Nonzero summands here correspond to the $n$-tuples $\beta$
of the form
\betan
\beta_j=\lambda_{\sigma(j)}-\sigma(j)+1,\qquad j=1,\dots,d,
\end{equation*}
and
\betan
\beta_j=-\lambda_{\tau(j)}+\tau(j)-1,\qquad j=d+1,\dots,n,
\end{equation*}
where $\sigma$ is a permutation of $\{1,\dots,d\}$ and $\tau$
is a permutation of
$\{d+1,\dots,n\}$, and
$\lambda$ is a diagram containing $\mu$
such that $\lambda$ has $d$ boxes on the main diagonal.
Dividing both sides of the above relation by the Vandermonde
determinant, we get the desired expansion formula.
\end{proof}

Now we obtain a tableau presentation
of the coefficients in the expansion of $s_{\mu}(x)$;
cf. Theorem~\ref{thm:tabre}. We assume, as before, that
$\lambda$ and $\mu$ have the same number of boxes $d$ on their
main diagonals.
By a {\it dual hook $\lambda/\mu$-tableau\/}
$T$ we will mean a tableau obtained by filling in the boxes
of $\lambda/\mu$ with integers in the following way.
The entries in the first $d$ rows
strictly decrease along the rows and weakly decrease
down the columns, and all entries in row $i$
belong to the set $\{0,-1,\dots,i-\lambda_i+1\}$ for $i=1,\dots,d$;
the entries in the first $d$ columns
strictly increase down the columns and weakly increase
along the rows, and all entries in column $j$
belong to the set $\{1,2,\dots,\lambda'_j-j\}$ for
$j=1,\dots,d$. Then we define the corresponding dual flagged
Schur function $\mathfrak{p}si_{\lambda/\mu}(a)$ by the formula
\betan
\mathfrak{p}si_{\lambda/\mu}(a)=\sum_T \mathfrak{p}rod_{\alpha\in\lambda/\mu} a^{}_{T(\alpha)},
\end{equation*}
summed over the dual hook $\lambda/\mu$-tableaux $T$.

\begin{thm}\lambdabel{thm:invtabre}
We have the expansion of the Schur function
$s_{\mu}(x)$
\betan
s_{\mu}(x)=\sum_{\lambda}(-1)^{m(\lambda/\mu)}\,
\mathfrak{p}si_{\lambda/\mu}(a)\, \widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
summed over diagrams $\lambda$ which
contain $\mu$ and such that $\lambda$ has
$d$ boxes on the main diagonal, where
$m(\lambda/\mu)$ denotes the total number of boxes in
the diagram $\lambda/\mu$ in rows $1,\dots,d$.
\end{thm}

\begin{proof}
This is deduced from
Proposition~\ref{prop:inve} and the formulas for
the flagged Schur functions in \cite[8th~Variation]{m:sf}
exactly as in the proof of Theorem~\ref{thm:tabre}.
\end{proof}

\begin{cor}\lambdabel{cor:inext}
For the expansion of the hook Schur function we have
\betan
s_{(\alpha\,s|\,s\beta)}(x)=
\sum_{p,\, q\mathfrak{g}eqslant 0}(-1)^p\,
e_p(a^{}_0,a^{}_{-1},\dots,a^{}_{-\alpha-p+1})\,
e_q(a^{}_1,a^{}_2,\dots,a^{}_{\beta+q})\,
\widehat s^{}_{(\alpha+p\,s|\,s\beta+q)}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
\end{cor}

\betax\lambdabel{ex:sonee}
We have
\betan
s_{(1)}(x)=
\sum_{p,\, q\mathfrak{g}eqslant 0}(-1)^p\,
a^{}_0\, a^{}_{-1}\dots a^{}_{-p+1}\,
a^{}_1\, a^{}_2\dots a^{}_{q}\,
\widehat s^{}_{(p\,s|\,s q)}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
\end{example}

As with the flagged Schur functions $\varphi_{\lambda/\mu}(a)$,
we have the following factorization formula
\betaql{invwhsufa}
\mathfrak{p}si_{\lambda/\mu}(a)=(-1)^{m(\lambda/\mu)}\,
\mathfrak{p}si_{(\lambda/\mu)_-}(a)\,\mathfrak{p}si_{(\lambda'/\mu')_-}(a'),
\end{equation}
where $(\lambda/\mu)_-$ denotes the part of the skew diagram
$\lambda/\mu$ whose boxes lie in the rows $d+1,d+2,\dots$.
An alternative tableau presentation
for the polynomials $\mathfrak{p}si_{(\lambda/\mu)_-}(a)$
is implied by the formulas \cite[$(8.2)$]{m:sf},
\cite{w:fs}. By \eqref{invwhsufa}, this also gives
alternative formulas for the
coefficients in the expansion of $s_{\mu}(x)$.

\begin{cor}\lambdabel{cor:invanexpa}
We have the tableau presentation
\betan
\mathfrak{p}si_{(\lambda/\mu)_-}(a)=
\sum_T \mathfrak{p}rod_{\alpha\in(\lambda/\mu)_-} a^{}_{T(\alpha)},
\end{equation*}
where the sum is taken
over the $(\lambda/\mu)_-$-tableaux $T$
whose entries in row $i$ belong to the set
$\{1,2,\dots,i-\lambda_i\}$ for $i=d+1,d+2,\dots$,
and the entries weakly increase along the rows
and strictly increase down
the columns.
\end{cor}

Completing this section we note that
the canonical comultiplication on the ring $\Lambda$
is naturally extended to the comultiplication
\betan
\Delta:\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to \widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\otimes^{}_{\,\mathbb{Q}\tss[a]}
\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation*}
defined on the generators by
\betan
\Delta\big(p_k(x)\big)= p_k(x)\otimes 1+1\otimes p_k(x).
\end{equation*}
Hence, Proposition~\ref{prop:skelrp} can be interpreted
in terms of $\Delta$ as the following decomposition
of the image of the dual Schur function
\betan
\Delta\big(\widehat s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)=
\sum_{\mu}
\widehat s_{\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)
\otimes \widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\lambda,\,\mu}
c_{\lambda\mu}^{\,\,s\nu}(a)\, \widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
\otimes \widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}

\section{Dual Littlewood--Richardson
polynomials}
\lambdabel{sec:dlr}
\setcounter{equation}{0}

It was pointed out in \cite[Remark~3.3]{orv:fs} that
the ring of supersymmetric functions $\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a)$
is equipped with the comultiplication $\Delta$
such that
\betan
\Delta\big(p_k(x/y)\big)= p_k(x/y)\otimes 1+1\otimes p_k(x/y);
\end{equation*}
cf. \cite[Chapter~I]{m:sfh}. The isomorphism \eqref{isom}
allows us to transfer the comultiplication
to the ring of double symmetric functions $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
so that $\Delta$ is a $\mathbb{Q}\tss[a]$-linear ring homomorphism
\betan
\Delta:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\otimes^{}_{\,\mathbb{Q}\tss[a]} \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation*}
such that
\betan
\Delta\big(p_k(x{\tss|\hspace{-1.5pt}|\tss} a)\big)= p_k(x{\tss|\hspace{-1.5pt}|\tss} a)\otimes 1+1\otimes p_k(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}

\begin{defin}\lambdabel{def:dulrpol}
The {\it dual Littlewood--Richardson
polynomials\/}
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ are
defined as the coefficients in the expansion
\betan
\Delta\big(s_{\nu}(x{\tss|\hspace{-1.5pt}|\tss} a)\big)=\sum_{\lambda,\,\mu}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)
\otimes s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
Equivalently, these polynomials can be found
from the decomposition
\betaql{skewdec}
s_{\,s\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}
\vskip-1.3\baselineskip
\qed
\end{defin}

In order to verify the equivalence of the definitions,
note that by \cite[Remark~3.3]{orv:fs},
\betan
\Delta\big(s_{\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\big)=\sum_{\mu}
s_{\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)
\otimes s_{\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
The claim now follows by the application
of Proposition~\ref{prop:imisom}.

It is clear from the definition that the
polynomial $\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ is nonzero
only if the inequality $|\nu|\mathfrak{g}eqslant |\lambda|+|\mu|$ holds.
In this case
it is a homogeneous polynomial in the variables $a_i$
of degree $|\nu|-|\lambda|-|\mu|$. Moreover, in the
particular case $|\nu|=|\lambda|+|\mu|$ the constant
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ equals $c_{\lambda\mu}^{\,s\nu}$,
the Littlewood--Richardson coefficient.

\begin{cor}\lambdabel{cor:symlr}
We have the following symmetry property
\betan
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=
\widehat c_{\lambda'\mu'}^{\,\,s\nu^{\,s\mathfrak{p}rime}}(a').
\end{equation*}
\end{cor}

\begin{proof}
By Proposition~\ref{prop:imisom} and Definition~\ref{def:skdsf},
we have
\betan
s_{\,s\nu/\mu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
The desired relations now follow from the symmetry
property \eqref{symprsu}.
\end{proof}

We can now prove that the dual Littlewood--Richardson polynomials
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
introduced in Definition~\ref{def:dulrpol} describe
the multiplication rule for the dual Schur functions.

\begin{thm}\lambdabel{thm:proddsf}
We have
the expansion
\betan
\widehat s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\nu} \widehat c_{\lambda\mu}^{\,\,s\nu}(a)\, \widehat s_{\,s\nu}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
\end{thm}

\begin{proof}
We argue as in the proof of the
classical analogue of this result; see \cite[Chapter~I]{m:sfh}.
Applying Corollary~\ref{cor:ssduco} for the
families of variables
$x=x'\cup x''$ and $y=y^{\,s\mathfrak{p}rime}\cup y^{\,s\mathfrak{p}rime\mathfrak{p}rime}$ we get
\betan
\begin{aligned}
\sum_{\nu\in\mathcal{P}} s_{\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\nu}(z{\tss|\hspace{-1.5pt}|\tss} a)
{}&=\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}
\frac{1+y^{\,s\mathfrak{p}rime}_i\, z_j}{1-x'_i\, z_j}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}
\frac{1+y^{\,s\mathfrak{p}rime\mathfrak{p}rime}_i\, z_j}{1-x''_i\, z_j}\\
{}&=\sum_{\lambda,\,\mu\in\mathcal{P}} s_{\lambda}(x'/y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
\, \widehat s_{\lambda}(z{\tss|\hspace{-1.5pt}|\tss} a)
\, s_{\mu}(x''/y^{\,s\mathfrak{p}rime\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat s_{\mu}(z{\tss|\hspace{-1.5pt}|\tss} a).
\end{aligned}
\end{equation*}
On the other hand, an alternative expansion
of the sum on the left hand side is obtained
by using the relation
\betan
s_{\nu}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda\subseteq\nu}s_{\lambda}(x'/y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
\, s_{\nu/\lambda}(x''/y^{\,s\mathfrak{p}rime\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
=\sum_{\lambda,\,\mu}s_{\lambda}(x'/y^{\,s\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a)
\, \widehat c_{\lambda\mu}^{\,\,s\nu}(a)\,
s_{\mu}(x''/y^{\,s\mathfrak{p}rime\mathfrak{p}rime}{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
implied by the combinatorial formula \eqref{taba}.
Therefore, the required relation follows by
comparing the two expansions.
\end{proof}

An explicit formula for the polynomials
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ is provided by the following
corollary, where the $c_{\alpha\beta}^{\,s\mathfrak{g}amma}$ denote the classical
Littlewood--Richardson coefficients defined by the decomposition
of the product of the Schur functions
\betan
s_{\alpha}(x)\, s_{\beta}(x)=
\sum_{\mathfrak{g}amma} c_{\alpha\beta}^{\,s\mathfrak{g}amma}\, s_{\,s\mathfrak{g}amma}(x).
\end{equation*}
We suppose that $\varphi_{\alpha/\lambda}(a)=\mathfrak{p}si_{\alpha/\lambda}(a)=0$
unless $\lambda\subseteq\alpha$ and the diagrams $\lambda$ and $\alpha$ have
the same number of boxes on their main diagonals.

\begin{cor}\lambdabel{cor:dulr}
We have
\betan
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=\sum_{\alpha,\beta,\mathfrak{g}amma}
(-1)^{n(\alpha/\lambda)+n(\beta/\mu)+m(\nu/\mathfrak{g}amma)}\,
c_{\alpha\beta}^{\,s\mathfrak{g}amma}\,
\varphi_{\alpha/\lambda}(a)\, \varphi_{\beta/\mu}(a)\,
\mathfrak{p}si_{\nu/\mathfrak{g}amma}(a),
\end{equation*}
summed over diagrams $\alpha$, $\beta$, $\mathfrak{g}amma$.
In particular,
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=0$ unless $\lambda\subseteq\nu$
and $\mu\subseteq\nu$.
\end{cor}

\begin{proof}
The formula follows from Theorems~\ref{thm:tabre}, \ref{thm:invtabre}
and \ref{thm:proddsf}. The second statement is implied
by the same property of the Littlewood--Richardson coefficients.
\end{proof}

\betax\lambdabel{ex:skewsf}
If $k\leqslant l$ and $k+l\leqslant m$ then
\betan
\widehat c_{(k)(l)}^{\,\,s(m)}(a)=\sum_{r+s=m-k-l}
(-1)^s\, h_r(a_0,a_{-1},\dots,a_{-k+1})
\, e_s(a_{-l},a_{-l-1},\dots,a_{-m+2}).
\end{equation*}
In particular,
\betan
\widehat c_{(1)(l)}^{\,\,s(m)}(a)=
(a_0-a_{-l})(a_0-a_{-l-1})\dots(a_0-a_{-m+2}).
\end{equation*}
Applying Corollary~\ref{cor:symlr}, we also get
\betan
\widehat c_{(1^k)(1^l)}^{\,\,s(1^m)}(a)=\sum_{r+s=m-k-l}
(-1)^r\, h_r(a_1,a_2,\dots,a_k)
\, e_s(a_{l+1},a_{l+2},\dots,a_{m-1})
\end{equation*}
and
\betan
\widehat c_{(1)(1^l)}^{\,\,s(1^m)}(a)=
(a_{l+1}-a_1)(a_{l+2}-a_1)\dots(a_{m-1}-a_1).
\end{equation*}
These relations provide explicit formulas for the images
of the double elementary and complete symmetric
functions $h_m(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $e_m(x{\tss|\hspace{-1.5pt}|\tss} a)$ with respect to
the comultiplication $\Delta$.
\qed
\end{example}

Another formula for the
dual Littlewood--Richardson polynomials
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$ can be obtained with the use
of the decomposition \eqref{skewdec}. We will consider the
skew double Schur function as the sequence of polynomials
$s_{\nu/\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)$ defined in \eqref{seqpol}.
For a given skew diagram $\nu/\mu$ consider the finite set
of variables $x=(x_1,\dots,x_n)$, where
$\nu'_j-\mu'_j\leqslant n$ for all $j$; that is,
the number of boxes in each column of $\nu/\mu$ does not
exceed $n$. Since the skew double Schur functions are
consistent with the evaluation homomorphisms \eqref{eval},
the polynomials $\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
are determined by the decomposition \eqref{skewdec},
where $x$ is understood as the above finite set of variables.

In order to formulate the result, introduce
$\nu/\mu$-{\it supertableaux\/} $T$
which are obtained by filling in the boxes of
$\nu/\mu$ with the symbols $1,1^{\,s\mathfrak{p}rime},\dots,n,n'$
in such a way that
in each row (resp. column) each primed
index is to the left (resp. above)
of each unprimed index;
unprimed indices weakly decrease along the rows and
strictly decrease down the columns;
primed indices strictly increase along the rows and
weakly increase down the columns.

Introduce the ordering on the set of boxes of
a skew diagram by reading them by columns from
left to right and from bottom to top in each column.
We call this the {\it column order\/}.
We shall write $\alpha\mathfrak{p}rec \beta$ if $\alpha$ (strictly) precedes $\beta$
with respect to the column order.

Suppose that $\lambda$ is a diagram. Given a sequence
of diagrams $R$ of the form
\betaql{r}
\varnothing=\rho^{(0)}\to\rho^{(1)}\to
\dots\to\rho^{(l-1)}\to\rho^{(l)}=\lambda,
\end{equation}
we let $r_i$ denote the row number of the box added to
the diagram $\rho^{(i-1)}$.
The sequence $r_1r_2\dots r_l$ is called the {\it Yamanouchi symbol\/}
of $R$.  Construct the set $\mathcal{T}(\nu/\mu,R)$ of {\it barred\/}
$\nu/\mu$-supertableaux
$T$ such that $T$ contains
boxes $\alpha_1,\dots,\alpha_l$ with
\betan
\alpha_1\mathfrak{p}rec\dots\mathfrak{p}rec\alpha_l\qquad\text{and}\qquad T(\alpha_i)=r_i,\quad
1\leqslant i\leqslant l,
\end{equation*}
where all entries $r_i$ are unprimed and the boxes
are listed in the column order
which is restricted to the
subtableau of $T$ formed by the unprimed indices.

We will distinguish the entries in $\alpha_1,\dots,\alpha_l$ by barring
each of them. So, an element of $\mathcal{T}(\nu/\mu,R)$ is a pair consisting
of a $\nu/\mu$-supertableau and a chosen sequence of barred entries
compatible with $R$.
We shall keep the notation $T$ for such a pair.

For each box $\alphapha$ with $\alpha_i\mathfrak{p}rec\alpha\mathfrak{p}rec\alpha_{i+1}$,
$0\leqslant i\leqslant l$,
which is occupied by an unprimed index,
set $\rho(\alphapha)=\rho^{(i)}$.

\begin{thm}\lambdabel{thm:supt}
The dual Littlewood--Richardson polynomials
can be given by
\betan
\begin{aligned}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)=\sum_R
\sum_{T}
\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, unprimed,\, unbarred}}{\alphapha\in\nu/\mu}}
&\big(a^{}_{\,s T(\alphapha)-\rho(\alphapha)^{}_{T(\alpha)}}-
a^{}_{\,s T(\alphapha)-c(\alphapha)}\big)\\
{}\times{}\mathfrak{p}rod_{\underset{\scriptstyle T(\alphapha)
\text{\,\, primed}}{\alphapha\in\nu/\mu}}
&\big(a^{}_{\,s T(\alphapha)-c(\alphapha)}
-a^{}_{\,s T(\alphapha)}\big),
\end{aligned}
\end{equation*}
summed over sequences $R$ of the form \eqref{r} and
barred supertableaux $T\in\mathcal{T}(\nu/\mu,R)$.
\end{thm}

\begin{proof}
Due to \eqref{seqpol}, we have
\betan
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)
=\sum_{\mu\subseteq\,\rho\,\subseteq\,s\nu}
\widetilde c_{\lambda\rho}^{\,\,s\nu}(a)\,
s_{\rho^{\,s\mathfrak{p}rime}/\mu^{\,s\mathfrak{p}rime}}(-a^{(n)} \,s|\,s {-}a),
\end{equation*}
where the polynomials $\widetilde c_{\lambda\rho}^{\,\,s\nu}(a)$
are defined by the decomposition
\betan
\widetilde s_{\nu/\rho}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda}
\widetilde c_{\lambda\rho}^{\,\,s\nu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
The desired formula is now implied by
\cite[Lemma~2.4]{m:lr} which gives the combinatorial
expression for the coefficients
$\widetilde c_{\lambda\rho}^{\,\,s\nu}(a)$ and thus takes care
of the unprimed part of $T$; the expression for the primed part
is implied by \eqref{facsf}.
\end{proof}

\begin{remark}\lambdabel{rem:open}
Both the formulas for $\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
provided by Corollary~\ref{cor:dulr}
and Theorem~\ref{thm:supt}
involve some terms which cancel pairwise.
It would be interesting to find a combinatorial presentation
of the polynomials $\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
analogous to \cite{kt:pe}, \cite{k:elr} or \cite{m:lr}
and to understand their positivity properties.
A possible way to find such a presentation could rely on
the vanishing theorem of the supersymmetric Schur functions
obtained in \cite[Theorems~5.1 \& 5.2]{orv:fs}; see also
\cite[Theorem~4.4]{m:fs} for a similar result.
\qed
\end{remark}

\betax\lambdabel{ex:superca}
In order to calculate the polynomial
$\widehat c_{(1)\,(2)}^{\,\,s(2^2)}(a)$, take
$\lambda=(1)$, $\mu=(2)$, $\nu=(2^2)$ and
$n=1$. The barred
supertableaux compatible with the sequence $\varnothing\to (1)$
are

\setlength{\unitlength}{0.75em}
\betagin{center}
\betagin{picture}(30,4.2)

\mathfrak{p}ut(0,0){\line(0,1){3.8}}
\mathfrak{p}ut(2,0){\line(0,1){3.8}}
\mathfrak{p}ut(4,0){\line(0,1){3.8}}
\mathfrak{p}ut(0,0){\line(1,0){4}}
\mathfrak{p}ut(0,2){\line(1,0){4}}

\mathfrak{p}ut(0.7,0.5){$\overline{1}$}
\mathfrak{p}ut(0.7,2.5){}
\mathfrak{p}ut(2.7,2.5){}
\mathfrak{p}ut(2.7,0.5){1}
\mathfrak{p}ut(0,2.2){\line(1,0){4}}
\mathfrak{p}ut(0,2.4){\line(1,0){4}}
\mathfrak{p}ut(0,2.6){\line(1,0){4}}
\mathfrak{p}ut(0,2.8){\line(1,0){4}}
\mathfrak{p}ut(0,3.0){\line(1,0){4}}
\mathfrak{p}ut(0,3.2){\line(1,0){4}}
\mathfrak{p}ut(0,3.4){\line(1,0){4}}
\mathfrak{p}ut(0,3.6){\line(1,0){4}}
\mathfrak{p}ut(0,3.8){\line(1,0){4}}

\mathfrak{p}ut(12,0){\line(0,1){3.8}}
\mathfrak{p}ut(14,0){\line(0,1){3.8}}
\mathfrak{p}ut(16,0){\line(0,1){3.8}}
\mathfrak{p}ut(12,0){\line(1,0){4}}
\mathfrak{p}ut(12,2){\line(1,0){4}}

\mathfrak{p}ut(12.7,0.5){1}
\mathfrak{p}ut(12.7,2.5){}
\mathfrak{p}ut(14.7,2.5){}
\mathfrak{p}ut(14.7,0.5){$\overline{1}$}
\mathfrak{p}ut(12,2.2){\line(1,0){4}}
\mathfrak{p}ut(12,2.4){\line(1,0){4}}
\mathfrak{p}ut(12,2.6){\line(1,0){4}}
\mathfrak{p}ut(12,2.8){\line(1,0){4}}
\mathfrak{p}ut(12,3.0){\line(1,0){4}}
\mathfrak{p}ut(12,3.2){\line(1,0){4}}
\mathfrak{p}ut(12,3.4){\line(1,0){4}}
\mathfrak{p}ut(12,3.6){\line(1,0){4}}
\mathfrak{p}ut(12,3.8){\line(1,0){4}}

\mathfrak{p}ut(24,0){\line(0,1){3.8}}
\mathfrak{p}ut(26,0){\line(0,1){3.8}}
\mathfrak{p}ut(28,0){\line(0,1){3.8}}
\mathfrak{p}ut(24,0){\line(1,0){4}}
\mathfrak{p}ut(24,2){\line(1,0){4}}

\mathfrak{p}ut(24.5,0.5){$1^{\,s\mathfrak{p}rime}$}
\mathfrak{p}ut(24.7,2.5){}
\mathfrak{p}ut(26.7,2.5){}
\mathfrak{p}ut(26.7,0.5){$\overline{1}$}
\mathfrak{p}ut(24,2.2){\line(1,0){4}}
\mathfrak{p}ut(24,2.4){\line(1,0){4}}
\mathfrak{p}ut(24,2.6){\line(1,0){4}}
\mathfrak{p}ut(24,2.8){\line(1,0){4}}
\mathfrak{p}ut(24,3.0){\line(1,0){4}}
\mathfrak{p}ut(24,3.2){\line(1,0){4}}
\mathfrak{p}ut(24,3.4){\line(1,0){4}}
\mathfrak{p}ut(24,3.6){\line(1,0){4}}
\mathfrak{p}ut(24,3.8){\line(1,0){4}}

\end{picture}
\end{center}
\setlength{\unitlength}{1pt}

\noindent
so that
\betan
\widehat c_{(1)\,(2)}^{\,\,s(2^2)}(a)
=a_0-a_1+a_1-a_2
+a_{2}-a_1=a_0-a_1.
\end{equation*}
Alternatively, we can take
$\lambda=(2)$, $\mu=(1)$, $\nu=(2^2)$ and
$n=2$. The barred
supertableaux compatible with the sequence $\varnothing\to (1)\to (2)$
are

\setlength{\unitlength}{0.75em}
\betagin{center}
\betagin{picture}(30,4.6)

\mathfrak{p}ut(0,0){\line(0,1){4}}
\mathfrak{p}ut(2,0){\line(0,1){4}}
\mathfrak{p}ut(4,0){\line(0,1){4}}
\mathfrak{p}ut(0,0){\line(1,0){4}}
\mathfrak{p}ut(0,2){\line(1,0){4}}
\mathfrak{p}ut(0,4){\line(1,0){4}}
\mathfrak{p}ut(0.7,0.5){$\overline{1}$}
\mathfrak{p}ut(0.7,2.5){}
\mathfrak{p}ut(2.7,2.5){2}
\mathfrak{p}ut(2.7,0.5){$\overline{1}$}
\mathfrak{p}ut(0,2.2){\line(1,0){2}}
\mathfrak{p}ut(0,2.4){\line(1,0){2}}
\mathfrak{p}ut(0,2.6){\line(1,0){2}}
\mathfrak{p}ut(0,2.8){\line(1,0){2}}
\mathfrak{p}ut(0,3.0){\line(1,0){2}}
\mathfrak{p}ut(0,3.2){\line(1,0){2}}
\mathfrak{p}ut(0,3.4){\line(1,0){2}}
\mathfrak{p}ut(0,3.6){\line(1,0){2}}
\mathfrak{p}ut(0,3.8){\line(1,0){2}}

\mathfrak{p}ut(12,0){\line(0,1){4}}
\mathfrak{p}ut(14,0){\line(0,1){4}}
\mathfrak{p}ut(16,0){\line(0,1){4}}
\mathfrak{p}ut(12,0){\line(1,0){4}}
\mathfrak{p}ut(12,2){\line(1,0){4}}
\mathfrak{p}ut(12,4){\line(1,0){4}}
\mathfrak{p}ut(12.7,0.5){$\overline{1}$}
\mathfrak{p}ut(12.7,2.5){}
\mathfrak{p}ut(14.7,2.5){$1^{\,s\mathfrak{p}rime}$}
\mathfrak{p}ut(14.7,0.5){$\overline{1}$}
\mathfrak{p}ut(12,2.2){\line(1,0){2}}
\mathfrak{p}ut(12,2.4){\line(1,0){2}}
\mathfrak{p}ut(12,2.6){\line(1,0){2}}
\mathfrak{p}ut(12,2.8){\line(1,0){2}}
\mathfrak{p}ut(12,3.0){\line(1,0){2}}
\mathfrak{p}ut(12,3.2){\line(1,0){2}}
\mathfrak{p}ut(12,3.4){\line(1,0){2}}
\mathfrak{p}ut(12,3.6){\line(1,0){2}}
\mathfrak{p}ut(12,3.8){\line(1,0){2}}

\mathfrak{p}ut(24,0){\line(0,1){4}}
\mathfrak{p}ut(26,0){\line(0,1){4}}
\mathfrak{p}ut(28,0){\line(0,1){4}}
\mathfrak{p}ut(24,0){\line(1,0){4}}
\mathfrak{p}ut(24,2){\line(1,0){4}}
\mathfrak{p}ut(24,4){\line(1,0){4}}
\mathfrak{p}ut(24.5,0.5){$\overline{1}$}
\mathfrak{p}ut(24.7,2.5){}
\mathfrak{p}ut(26.7,2.5){$2^{\,s\mathfrak{p}rime}$}
\mathfrak{p}ut(26.7,0.5){$\overline{1}$}
\mathfrak{p}ut(24,2.2){\line(1,0){2}}
\mathfrak{p}ut(24,2.4){\line(1,0){2}}
\mathfrak{p}ut(24,2.6){\line(1,0){2}}
\mathfrak{p}ut(24,2.8){\line(1,0){2}}
\mathfrak{p}ut(24,3.0){\line(1,0){2}}
\mathfrak{p}ut(24,3.2){\line(1,0){2}}
\mathfrak{p}ut(24,3.4){\line(1,0){2}}
\mathfrak{p}ut(24,3.6){\line(1,0){2}}
\mathfrak{p}ut(24,3.8){\line(1,0){2}}

\end{picture}
\end{center}
\setlength{\unitlength}{1pt}

\noindent
so that
\betan
\widehat c_{(1)\,(2)}^{\,\,s(2^2)}(a)
=a_2-a_1+a_0-a_1
+a_{1}-a_2=a_0-a_1.
\end{equation*}
This agrees with the previous calculation
and the formula implied by Corollary~\ref{cor:dulr}.
\end{example}

\betax\lambdabel{ex:supediff}
Theorem~\ref{thm:supt} gives formulas
for the polynomials $\widehat c_{(k)(l)}^{\,\,s(m)}(a)$
and $\widehat c_{(1^k)(1^l)}^{\,\,s(1^m)}(a)$
in a different form
as compared to Example~\ref{ex:skewsf}.
If $k+l\leqslant m$ then
\betan
\begin{aligned}
\widehat c_{(k)(l)}^{\,\,s(m)}(a)=\sum
&(a_0-a_{-l})(a_0-a_{-l-1})\cdots (a_0-a_{-l-i_1+1})\\
&{}\times{}(a_{-1}-a_{-l-i_1-1})\cdots (a_{-1}-a_{-l-i_2+1})\\
&{}\times{}\cdots
(a_{-k+1}-a_{-l-i_{k-1}-1})\cdots (a_{-k+1}-a_{-m+2})
\end{aligned}
\end{equation*}
summed over the sets of indices
$0\leqslant i_1<\dots<i_{k-1}\leqslant m-l-2$.
A similar expression for $\widehat c_{(1^k)(1^l)}^{\,\,s(1^m)}(a)$
follows by the application of Corollary~\ref{cor:symlr}.
\end{example}

\section{Transition matrices}
\lambdabel{sec:ome}
\setcounter{equation}{0}

\subsection{Pairing between the double and dual symmetric
functions}

We now prove alternative expansion formulas for the infinite product
which occurs in the Cauchy formula \eqref{cchid}.
These formulas turn into the well known identities
when $a$ is specialized to the sequence of zeros; see
\cite[Chapter~I]{m:sfh}.

Let $\lambda=(\lambda_1,\dots,\lambda_l)$ be a partition and suppose
that the length of $\lambda$ does not
exceed $l$. Using the notation
\eqref{dumon}, introduce the
{\it dual monomial symmetric function\/}
$\widehat m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ by the formula
\betan
\widehat m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=
\sum_{\sigma} (x_{\sigma(1)},a)^{\lambda_1}\, (x_{\sigma(2)},a)^{\lambda_2}\dots
(x_{\sigma(l)},a)^{\lambda_l},
\end{equation*}
summed over permutations $\sigma$ of the $x_i$
which give distinct monomials.

For a partition $\lambda=(1^{m_1}\, 2^{m_2}\dots)$ set
$z_{\lambda}=\mathfrak{p}rod_{i\mathfrak{g}eqslant 1} i^{m_i}\, m_i!$.

\begin{prop}\lambdabel{prop:expa}
We have
the expansions
\betaql{hmde}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}}
h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat m_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation}
and
\betaql{ppde}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}}
z_{\lambda}^{-1}\, p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, p_{\lambda}(y).
\end{equation}
\end{prop}

\begin{proof}
Let us set
\betan
H(t)=\mathfrak{p}rod_{i=1}^{\infty}\frac{1-a_i\,s t}{1-x_i\,s t}.
\end{equation*}
Then using \eqref{genh} and arguing as in
\cite[Chapter~I]{m:sfh}, we can write
\betan
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\mathfrak{p}rod_{j\mathfrak{g}eqslant 1} H(y_j)
=\mathfrak{p}rod_{j\mathfrak{g}eqslant 1}\sum_{k=0}^{\infty} h_k(x{\tss|\hspace{-1.5pt}|\tss} a)
\, (y_j,a)^k=\sum_{\lambda\in\mathcal{P}}
h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat m_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
which proves \eqref{hmde}. For the proof of \eqref{ppde}
note that
\betan
\begin{aligned}
\ln H(t)&=\sum_{i\mathfrak{g}eqslant 1}\Big(\ln(1-a_it)-\ln(1-x_it)\Big)\\
{}&=\sum_{i\mathfrak{g}eqslant 1}\sum_{k\mathfrak{g}eqslant 1}
\Big(\frac{x_i^k\, t^k}{k}-\frac{a_i^k\, t^k}{k}\Big)
=\sum_{k\mathfrak{g}eqslant 1}\frac{p_k(x{\tss|\hspace{-1.5pt}|\tss} a) t^k}{k}.
\end{aligned}
\end{equation*}
Hence,
\betan
H(t)=\sum_{\lambda\in\mathcal{P}} z_{\lambda}^{-1}\, p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, t^{|\lambda|}.
\end{equation*}
Now apply this relation to the sets of variables
$x$ and $a$ respectively replaced with the sets $\{x_iy_j\}$
and $\{a_iy_j\}$. Then $p_k(x{\tss|\hspace{-1.5pt}|\tss} a)$ is replaced by
$p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, p_{\lambda}(y)$, and \eqref{ppde}
follows by putting $t=1$.
\end{proof}

Now define the $\mathbb{Q}\tss[a]$-bilinear pairing
between the rings $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$,
\betaql{pairi}
\lambdangle\ ,\ \rangle:\big(\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)\big)\to \mathbb{Q}\tss[a],
\end{equation}
by setting
\betaql{defpa}
\big\lambdangle h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat m_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu}.
\end{equation}
Clearly, $\lambdangle u,\widehat v\rangle$ is
a well-defined polynomial in $a$ for any elements
$u\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat v\in\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$ which is determined
from \eqref{defpa} by linearity.

The following is an analogue of the duality properties
of the classical bases of the ring of symmetric functions;
see \cite[Chapter~I]{m:sfh}.

\begin{prop}\lambdabel{prop:dubas}
Let $\{u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\}$
and $\{\widehat v_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\}$ be families of elements
of rings $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$, respectively,
which are parameterized by
all partitions. Suppose that for any $n\mathfrak{g}eqslant 0$
the highest degree components in $x$ {\rm(}resp.,
the lowest degree components in $y${\rm)}
of the elements $u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ {\rm(}resp.,
$\widehat v_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)${\rm)}
with $|\lambda|=n$
form a basis of the space of homogeneous symmetric
functions in $x$ {\rm(}resp., $y${\rm)} of degree $n$.
Then the following conditions are equivalent:
\betaql{uvdu}
\big\lambdangle u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat v_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu},\qquad\text{for all}\quad \lambda,\mu;
\end{equation}
\betaql{unsum}
\sum_{\lambda\in\mathcal{P}}
u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat v_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)
=\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}.
\end{equation}
\end{prop}

\begin{proof}
We only need to slightly modify the respective argument of
\cite[Chapter~I]{m:sfh}.
Write
\betan
u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\rho} A_{\lambda\rho}(a)\, h_{\rho}(x{\tss|\hspace{-1.5pt}|\tss} a),
\qquad
\widehat v_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\sigma} B_{\mu\sigma}(a)\, \widehat m_{\sigma}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
where the first sum is taken over partitions $\rho$
with $|\rho|\leqslant |\lambda|$, while the second
is taken over partitions $\sigma$
with $|\sigma|\mathfrak{g}eqslant |\mu|$.
Then
\betan
\big\lambdangle u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat v_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\sum_{\rho} A_{\lambda\rho}(a)\, B_{\mu\rho}(a).
\end{equation*}
Hence, condition \eqref{uvdu} is equivalent to
\betaql{abmu}
\sum_{\rho} A_{\lambda\rho}(a)\, B_{\mu\rho}(a)=\delta^{}_{\lambda\mu}.
\end{equation}
On the other hand, due to \eqref{hmde},
\eqref{unsum} can be written as
\betan
\sum_{\lambda\in\mathcal{P}}
u_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat v_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\rho\in\mathcal{P}}
h_{\rho}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat m_{\rho}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
which is equivalent to
\betan
\sum_{\lambda} A_{\lambda\rho}(a)\, B_{\lambda\sigma}(a)=\delta^{}_{\rho\sigma}.
\end{equation*}
This condition is easily verified
to be equivalent to \eqref{abmu}.
\end{proof}

Applying Theorem~\ref{thm:cch} and Proposition~\ref{prop:expa} we get
the following corollary.

\begin{cor}\lambdabel{cor:dumup}
Under the pairing \eqref{pairi} we have
\betan
\big\lambdangle s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat s_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu}
\qquad\text{and}\qquad
\big\lambdangle p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),p_{\mu}(y)\big\rangle
=\delta^{}_{\lambda\mu}\, z_{\lambda}.
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{cor}

Thus, the symmetric functions $\widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)$
are dual to the double Schur functions
$s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$
in sense of the pairing \eqref{pairi}.

Using the isomorphism \eqref{isom} and the pairing
\eqref{pairi}, we get another
$\mathbb{Q}\tss[a]$-bilinear pairing
\betaql{pairisu}
\lambdangle\ ,\ \rangle:\big(\Lambda(x/y{\tss|\hspace{-1.5pt}|\tss} a),\widehat\Lambda(z{\tss|\hspace{-1.5pt}|\tss} a)\big)\to \mathbb{Q}\tss[a]
\end{equation}
such that
\betaql{duoda}
\big\lambdangle s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a),\widehat s_{\mu}(z{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu}.
\end{equation}
Note that Proposition~\ref{prop:dubas} can be easily
reformulated for the pairing \eqref{pairisu}. In particular,
the condition \eqref{unsum} is now replaced by
\betaql{unsumsu}
\sum_{\lambda\in\mathcal{P}}
u_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat v_{\lambda}(z{\tss|\hspace{-1.5pt}|\tss} a)
=\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+y_i\, z_j}{1-x_i\, z_j}.
\end{equation}
This implies that
\betaql{duod}
\big\lambdangle s_{\lambda}(x/y),s_{\mu}(z)\big\rangle
=\delta^{}_{\lambda\mu},
\end{equation}
where $s_{\lambda}(x/y)$ denotes the ordinary supersymmetric
Schur function which is obtained from $s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$
by the specialization $a_i=0$.
Together with Theorems~\ref{thm:tabre} and
\ref{thm:invtabre}, the relations
\eqref{duoda} and \eqref{duod} imply the following
expansions for the supersymmetric Schur functions.

\begin{cor}\lambdabel{cor:expasdo}
We have the decompositions
\betan
s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu}(-1)^{m(\lambda/\mu)}\,
\mathfrak{p}si_{\lambda/\mu}(a)\, s_{\mu}(x/y),
\end{equation*}
summed over diagrams $\mu$ contained
in $\lambda$ and such that $\lambda$ and $\mu$ have
the same number of
boxes on the main diagonal; and
\betan
s_{\mu}(x/y)=\sum_{\lambda}(-1)^{n(\lambda/\mu)}\,
\varphi_{\lambda/\mu}(a)\, s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
summed over diagrams $\lambda$ which
contain $\mu$ and such that $\lambda$ and $\mu$ have
the same number of
boxes on the main diagonal.
\qed
\end{cor}

Note that expressions for $\mathfrak{p}si_{\lambda/\mu}(a)$ and
$\varphi_{\lambda/\mu}(a)$ in terms of determinants as
in Propositions~\ref{prop:dsfexp} and \ref{prop:inve}
were given in \cite{orv:fs}.
Corollary~\ref{cor:expasdo}
gives new tableau formulas for these coefficients.
Moreover, under the specialization $a_i=-i+1/2$
the supersymmetric Schur functions
$s_{\lambda}(x/y{\tss|\hspace{-1.5pt}|\tss} a)$ turn into the {\it Frobenius--Schur
functions\/} $Fs_{\mu}$; see \cite{orv:fs}.
Hence, the transition coefficients between the
$Fs_{\mu}$ and the Schur functions can be
found as follows; cf. \cite[Theorem~2.6]{orv:fs}.

\begin{cor}\lambdabel{cor:frschur}
We have the decompositions
\betan
Fs_{\lambda}=\sum_{\mu}(-1)^{m(\lambda/\mu)}\,
\mathfrak{p}si_{\lambda/\mu}\, s_{\mu}(x/y)
\end{equation*}
and
\betan
s_{\mu}(x/y)=\sum_{\lambda}(-1)^{n(\lambda/\mu)}\,
\varphi_{\lambda/\mu}\, Fs_{\lambda},
\end{equation*}
where $\mathfrak{p}si_{\lambda/\mu}$
and $\varphi_{\lambda/\mu}$ are the respective values of
the polynomials $\mathfrak{p}si_{\lambda/\mu}(a)$ and
$\varphi_{\lambda/\mu}(a)$ at $a_i=-i+1/2$,
$i\in\mathbb{Z}\tss$.
\qed
\end{cor}

Using the notation of Corollary~\ref{cor:expasdo} and
applying the isomorphism \eqref{isom} we get the respective
expansion formulas involving the double Schur functions.

\begin{cor}\lambdabel{cor:expasdouble}
We have the decompositions
\betan
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu}(-1)^{m(\lambda/\mu)}\,
\mathfrak{p}si_{\lambda/\mu}(a)\, s_{\mu}(x/{-}a^+)
\end{equation*}
and
\betan
s_{\mu}(x/{-}a^+)=\sum_{\lambda}(-1)^{n(\lambda/\mu)}\,
\varphi_{\lambda/\mu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
where $a^+=(a_1,a_2,\dots)$.
\qed
\end{cor}

Expressions for the coefficients
in the expansions relating the double and ordinary
Schur functions or polynomials can be found
in \cite{k:elr}, \cite{k:pf}, \cite{m:sf},
\cite{m:lr} and \cite{ms:lr}.

Let us now recall
the isomorphism $\omega^{}_a:\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$ and
the involution $\widehat\omega^{}:\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)\to \widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$;
see \eqref{omega} and \eqref{whome}.
Since every polynomial $c(a)\in \mathbb{Q}\tss[a]$ can be regarded
as an element of $\mathbb{Q}\tss[a']$, the ring $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$
can be naturally identified with $\widehat\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
via the map $c(a)\mapsto c^{\,s\mathfrak{p}rime}(a')$, where
$c^{\,s\mathfrak{p}rime}(a')=c(a)$
as polynomials in the $a_i$, $i\in\mathbb{Z}\tss$.

\begin{prop}\lambdabel{prop:oma}
For any elements $u\in \Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ and $\widehat v\in \widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$
we have
\betan
\big\lambdangle \omega^{}_a\,s u,\,\widehat\omega^{}\,s \widehat v\big\rangle'
=\big\lambdangle u, \widehat v\big\rangle,
\end{equation*}
where $\lambdangle \ ,\ \rangle'$ denotes the pairing
\eqref{pairi} between $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a')$
and $\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)\sigmameq \widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a')$.
\end{prop}

\begin{proof}
It suffices to take $u=s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)$ and
$\widehat v=\widehat s_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)$.
Using \eqref{omsla} and \eqref{whome}, we get
\betan
\big\lambdangle \omega^{}_a\,s s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\,\widehat\omega^{}\,s \widehat s_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle'
=\big\lambdangle s_{\lambda'}(x{\tss|\hspace{-1.5pt}|\tss} a'),
\, \widehat s_{\mu'}(y{\tss|\hspace{-1.5pt}|\tss} a')\big\rangle'.
\end{equation*}
By Corollary~\ref{cor:dumup} this equals $\delta^{}_{\lambda\mu}$,
and hence coincides with
$\lambdangle s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\, \widehat s_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\rangle$.
\end{proof}

Introduce the {\it dual forgotten symmetric functions\/}
$\widehat f_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)\in\widehat\Lambda(y{\tss|\hspace{-1.5pt}|\tss} a)$ as the images
of the dual monomial symmetric functions
under the involution $\widehat\omega^{}$, that is,
\betan
\widehat f_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)=\widehat\omega^{}\, \widehat m_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a'),\qquad \lambda\in\mathcal{P}.
\end{equation*}
Furthermore,
for any partition $\lambda$ define the {\it double
monomial symmetric functions\/} $m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
and the {\it double forgotten symmetric functions\/}
$f_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\in\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$ by the relations
\betaql{hmdua}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}}
m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat h_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)
\end{equation}
and
\betaql{ppdua}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1-a_i\, y_j}{1-x_i\, y_j}
=\sum_{\lambda\in\mathcal{P}}
f_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat e_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation}

Hence, by Proposition~\ref{prop:dubas},
under the pairing \eqref{pairi}
we have
\betaql{pamh}
\big\lambdangle m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat h_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu}
\qquad\text{and}\qquad
\big\lambdangle f_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),\widehat e_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)\big\rangle
=\delta^{}_{\lambda\mu}.
\end{equation}
Moreover, Propositions~\ref{prop:om} and \ref{prop:oma} imply
\betan
\omega^{}_a: m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto h_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a'),
\qquad f_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a'),
\qquad p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\mapsto \varepsilon_{\lambda}\, p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a'),
\end{equation*}
where $\varepsilon_{\lambda}=(-1)^{|\lambda|-\ell(\lambda)}$. To check the latter
relation we need to recall that under
the involution $\omega^{}$ of the ring of symmetric functions we have
$\omega^{}:p_{\lambda}(y)\mapsto
\varepsilon_{\lambda}\, p_{\lambda}(y)$; see \cite[Chapter~I]{m:sfh}.

We can now obtain
analogues of the decomposition of Corollary~\ref{cor:duco}
for other families of symmetric functions.

\begin{cor}\lambdabel{cor:decmf}
We have the decompositions
\betan
\begin{aligned}
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+x_i\, y_j}{1+a_i\, y_j}
&=\sum_{\lambda\in\mathcal{P}} e_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\,
\widehat m_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a^{\,\mathfrak{p}rime}),\\[1em]
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+x_i\, y_j}{1+a_i\, y_j}
&=\sum_{\lambda\in\mathcal{P}} \varepsilon_{\lambda}\, z_{\lambda}^{-1}\,
p_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, p_{\lambda}(y),\\[1em]
\mathfrak{p}rod_{i,\, j\mathfrak{g}eqslant 1}\frac{1+x_i\, y_j}{1+a_i\, y_j}
&=\sum_{\lambda\in\mathcal{P}} m_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)\, \widehat e_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a^{\,\mathfrak{p}rime}).
\end{aligned}
\end{equation*}
\end{cor}

\begin{proof}
The relations follow by the application of $\omega^{}_a$
to the expansions \eqref{hmde}, \eqref{ppde} and
by the application of $\widehat\omega^{}$ to \eqref{hmdua}.
\end{proof}

Note that relations of this kind involving the forgotten
symmetric functions can be obtained in a similar way.

\subsection{Kostka-type and character polynomials}

The entries of the transition matrices between the classical
bases of the ring of symmetric functions
can be expressed in terms of the Kostka numbers $K_{\lambda\mu}$
and the values ${\rm{ch}\,}i^{\lambda}_{\mu}$ of the irreducible
characters of the symmetric groups; see \cite[Sections~I.6,~I.7]{m:sfh}.
By analogy with the classical case, introduce the
{\it Kostka-type polynomials\/} $K_{\lambda\mu}(a)$ and
the {\it character polynomials\/} ${\rm{ch}\,}i^{\lambda}_{\mu}(a)$
as well as their dual counterparts
$\widehat K_{\lambda\mu}(a)$ and $\widehat{\rm{ch}\,}i^{\,\lambda}_{\mu}(a)$
by the respective expansions
\betan
s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu} K_{\lambda\mu}(a)\, m_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a),
\qquad
\widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\mu}
\widehat K_{\lambda\mu}(a)\, \widehat m_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a),
\end{equation*}
and
\betan
p_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda} {\rm{ch}\,}i^{\lambda}_{\mu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\qquad
p_{\mu}(y)=\sum_{\lambda} \widehat{\rm{ch}\,}i^{\,\lambda}_{\mu}(a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}
If $|\lambda|=|\mu|$, then
\betaql{cla}
K_{\lambda\mu}(a)=\widehat K_{\lambda\mu}(a)=K_{\lambda\mu}\qquad\text{and}\qquad
{\rm{ch}\,}i^{\lambda}_{\mu}(a)=\widehat{\rm{ch}\,}i^{\,\lambda}_{\mu}(a)={\rm{ch}\,}i^{\lambda}_{\mu}.
\end{equation}
Moreover, $K_{\lambda\mu}(a)$ and $\widehat{\rm{ch}\,}i^{\,\lambda}_{\mu}(a)$
are zero unless $|\lambda|\mathfrak{g}eqslant |\mu|$, while
$\widehat K_{\lambda\mu}(a)$ and ${\rm{ch}\,}i^{\lambda}_{\mu}(a)$
are zero unless $|\lambda|\leqslant |\mu|$.

Using the duality properties of the double and dual symmetric
functions, we can get all other transition matrices
in the same way as this is done in \cite[Sections~I.6,~I.7]{m:sfh}.
In particular, we have the relations
\betan
h_{\mu}(x{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda} \widehat K_{\lambda\mu}(a)\, s_{\lambda}(x{\tss|\hspace{-1.5pt}|\tss} a),
\qquad
\widehat h_{\mu}(y{\tss|\hspace{-1.5pt}|\tss} a)=\sum_{\lambda} K_{\lambda\mu}(a)\, \widehat s_{\lambda}(y{\tss|\hspace{-1.5pt}|\tss} a).
\end{equation*}

The Littlewood--Richardson
polynomials $c_{\lambda\mu}^{\,s\nu}(a)$ defined in \eqref{lrpoldef}
are Graham positive
as they can be written as
polynomials in the differences $a_i-a_j$, $i<j$, with
positive integer coefficients; see \cite{g:pe}.
Explicit positive formulas for
$c_{\lambda\mu}^{\,\nu}(a)$
were found in \cite{kt:pe}, \cite{k:elr} and \cite{m:lr}.
Using the fact that $h_k(x{\tss|\hspace{-1.5pt}|\tss} a)$ coincides with
$s_{(k)}(x{\tss|\hspace{-1.5pt}|\tss} a)$, we come to the following
expression for the polynomials $\widehat K_{\lambda\mu}(a)$:
\betan
\widehat K_{\lambda\mu}(a)=\sum_{\rho^{(1)},\dots,\,\rho^{(l-2)}}
c_{(\mu_1)\,s\rho^{(1)}}^{\,s\lambda}(a)\,
c_{(\mu_2)\,s\rho^{(2)}}^{\,s\rho^{(1)}}(a)\,
\dots c_{(\mu_{l-2})\,s\rho^{(l-2)}}^{\,s\rho^{(l-3)}}(a)\,
c_{(\mu_{l-1})\,s(\mu_{l})}^{\,s\rho^{(l-2)}}(a),
\end{equation*}
summed over partitions $\rho^{(i)}$,
where $\mu=(\mu_1,\dots\mu_l)$. In particular, each
dual Kostka-type polynomial
$\widehat K_{\lambda\mu}(a)$ is Graham positive. For a more explicit
tableau presentation of the polynomials $\widehat K_{\lambda\mu}(a)$
see \cite{f:tm}.

\betax\lambdabel{ex:kona}
We have
\betan
\widehat K_{(3\, 2)\,s(3\, 2\, 1)}(a)=
\sum_{\rho}
c_{(3)\,s\rho}^{\,s(3\, 2)}(a)\,
c_{(2)\,s(1)}^{\,s\rho}(a)
=c_{(3)\,s(2)}^{\,s(3\, 2)}(a)\, c_{(2)\,s(1)}^{\,s(2)}(a)
+c_{(3)\,s(2\, 1)}^{\,s(3\, 2)}(a)
\, c_{(2)\,s(1)}^{\,s(2\, 1)}(a).
\end{equation*}
Now, $c_{(3)\,s(2)}^{\,s(3\, 2)}(a)=c_{(3)\,s(2)}^{\,s(3\, 2)}=1$
and $c_{(2)\,s(1)}^{\,s(2\, 1)}(a)=c_{(2)\,s(1)}^{\,s(2\, 1)}=1$,
while applying \cite[Theorem~2.1]{m:lr} we get
$c_{(2)\,s(1)}^{\,s(2)}(a)=a_{-1}-a_1$ and
$c_{(3)\,s(2\, 1)}^{\,s(3\, 2)}(a)=a_{-2}-a_2$. Hence,
\betan
\widehat K_{(3\, 2)\,s(3\, 2\, 1)}(a)=
a_{-2}+a_{-1}-a_1-a_2.
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{example}

The polynomials $K_{\lambda\mu}(a)$ can be calculated
by the following procedure. Given a partition
$\mu=(\mu_1,\dots,\mu_l)$,
write each dual complete
symmetric function $\widehat h_{\mu_i}(y{\tss|\hspace{-1.5pt}|\tss} a)$ as a series of the hook
Schur functions with coefficients in $\mathbb{Q}\tss[a]$ using
Corollary~\ref{cor:sone}. Then multiply the Schur
functions using the classical Littlewood--Richardson rule.
Finally, use Theorem~\ref{thm:invtabre} to
represent each Schur function as a series of the
dual Schur functions.

\betax\lambdabel{ex:konadu}
By Example~\ref{ex:sone}, $\widehat h_1(y{\tss|\hspace{-1.5pt}|\tss} a)^2$ equals
\betan
\big(s_{(1)}(y)+a_0\,s s_{(2)}(y)-
a_1\,s s_{(1^2)}(y)+a^2_0\,s s_{(3)}(y)
-a_0\,s a_1\,s s_{(2\, 1)}(y)+a^2_1\,s s_{(1^3)}(y)+\dots\big)^2.
\end{equation*}
Hence, multiplying the Schur functions, we find that
\betan
\begin{aligned}
\widehat h_1(y{\tss|\hspace{-1.5pt}|\tss} a)^2&=s_{(2)}(y)+s_{(1^2)}(y)+2\,s a_0\,s s_{(3)}(y)
+2\,s(a_0-a_1)\,s s_{(2\, 1)}(y)-2\,s a_1\,s s_{(1^3)}(y)\\
{}&+3\,s a^2_0\,s s_{(4)}(y)
+(3\,s a^2_0-4\,s a_0\,s a_1)\,s s_{(3\, 1)}(y)
+(a^2_0+a^2_1-2\,s a_0\,s a_1)\,s s_{(2^2)}(y)\\
{}&+(3\,s a^2_1-4\,s a_0\,s a_1)\,s s_{(2\, 1^2)}(y)
+3\,s a^2_1\,s s_{(1^4)}(y)+\cdots.
\end{aligned}
\end{equation*}
Expanding now each Schur function with the use of
Theorem~\ref{thm:invtabre} or Corollary~\ref{cor:inext},
we come to
\betan
\begin{aligned}
\widehat h_1(y{\tss|\hspace{-1.5pt}|\tss} a)^2&=\widehat s_{(2)}(y{\tss|\hspace{-1.5pt}|\tss} a)+\widehat s_{(1^2)}(y{\tss|\hspace{-1.5pt}|\tss} a)
+(a_0-a_{-1})\,s \widehat s_{(3)}(y{\tss|\hspace{-1.5pt}|\tss} a)
+(a_0-a_1)\,s \widehat s_{(2\, 1)}(y{\tss|\hspace{-1.5pt}|\tss} a)\\
{}&+(a_2-a_1)\,s \widehat s_{(1^3)}(y{\tss|\hspace{-1.5pt}|\tss} a)
+(a_0-a_{-2})\,s(a_0-a_{-1})\,s \widehat s_{(4)}(y{\tss|\hspace{-1.5pt}|\tss} a)\\
{}&+(a_0-a_1)\,s(a_0-a_{-1})\,s \widehat s_{(3\, 1)}(y{\tss|\hspace{-1.5pt}|\tss} a)
+(a_0-a_1)^2\,s  \widehat s_{(2^2)}(y{\tss|\hspace{-1.5pt}|\tss} a)\\
{}&+(a_1-a_0)\,s(a_1-a_2)\,s \widehat s_{(2\, 1^2)}(y{\tss|\hspace{-1.5pt}|\tss} a)
+(a_1-a_2)\,s(a_1-a_3)\,s \widehat s_{(1^4)}(y{\tss|\hspace{-1.5pt}|\tss} a)+\cdots,
\end{aligned}
\end{equation*}
thus calculating the first few polynomials $K_{\lambda\,s(1^2)}(a)$.
\qed
\end{example}

\betax\lambdabel{ex:monom}
Using Example~\ref{ex:konadu}, we can calculate the first few
double monomial symmetric functions:
\betan
\begin{aligned}
m_{(1)}(x{\tss|\hspace{-1.5pt}|\tss} a)&=s_{(1)}(x{\tss|\hspace{-1.5pt}|\tss} a),\qquad
m_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a)=s_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a)\\
m_{(2)}(x{\tss|\hspace{-1.5pt}|\tss} a)&=s_{(2)}(x{\tss|\hspace{-1.5pt}|\tss} a)-s_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a)\\
m_{(1^3)}(x{\tss|\hspace{-1.5pt}|\tss} a)&=s_{(1^3)}(x{\tss|\hspace{-1.5pt}|\tss} a)
+(a_1-a_2)\,s s_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a)\\
m_{(2\, 1)}(x{\tss|\hspace{-1.5pt}|\tss} a)&=s_{(2\, 1)}(x{\tss|\hspace{-1.5pt}|\tss} a)-2\,s s_{(1^3)}(x{\tss|\hspace{-1.5pt}|\tss} a)
+(2\,s a_2-a_1-a_0)\,s s_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a)\\
m_{(3)}(x{\tss|\hspace{-1.5pt}|\tss} a)&=s_{(3)}(x{\tss|\hspace{-1.5pt}|\tss} a)-
s_{(2\, 1)}(x{\tss|\hspace{-1.5pt}|\tss} a)+\,s s_{(1^3)}(x{\tss|\hspace{-1.5pt}|\tss} a)
+(a_{-1}-a_0)\,s s_{(1^2)}(x{\tss|\hspace{-1.5pt}|\tss} a).
\end{aligned}
\end{equation*}
\vskip-1.1\baselineskip
\qed
\end{example}

The following formula for the dual character
polynomials is implied by
Corollary~\ref{cor:expasdouble}.

\begin{cor}\lambdabel{cor:char}
We have
\betan
\widehat{\rm{ch}\,}i^{\,\lambda}_{\mu}(a)=\sum_{\rho}
(-1)^{m(\lambda/\rho)}\, {\rm{ch}\,}i^{\rho}_{\mu}\,
\mathfrak{p}si_{\lambda/\rho}(a),
\end{equation*}
summed over diagrams $\rho$ with $|\rho|=|\mu|$.
\qed
\end{cor}

\section{Interpolation formulas}
\lambdabel{sec:idd}
\setcounter{equation}{0}

\subsection{Rational expressions for the transition coefficients}

Applying Proposition~\ref{prop:interp}, we can get
expressions for the polynomials
$\widehat K_{\lambda\mu}(a)$, ${\rm{ch}\,}i_{\mu}^{\lambda}(a)$,
$\widehat c_{\lambda\mu}^{\,\,s\nu}(a)$
and $c_{\lambda\mu}^{\,s\nu}(a)$ as rational
functions in the variables $a_i$.

\begin{prop}\lambdabel{prop:ratfe}
We have the expressions
\betagin{align}
\lambdabel{kostka}
\widehat K_{\lambda\mu}(a)&=\sum_{R}\sum_{k=0}^{l}
\frac{h_{\mu}(a_{\rho^{(k)}}{\tss|\hspace{-1.5pt}|\tss} a)}{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(l)}}|)},\\
\lambdabel{charact}
{\rm{ch}\,}i_{\mu}^{\lambda}(a)&=\sum_{R}\sum_{k=0}^{l}
\frac{p_{\mu}(a_{\rho^{(k)}}{\tss|\hspace{-1.5pt}|\tss} a)}{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(l)}}|)},\\
\lambdabel{duallr}
\widehat c_{\lambda\mu}^{\,\,s\nu}(a)&=\sum_{R}\sum_{k=0}^{l}
\frac{s_{\nu/\mu}(a_{\rho^{(k)}}{\tss|\hspace{-1.5pt}|\tss} a)}{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(l)}}|)},
\end{align}
summed over all sequences of partitions $R$
of the form
\betan
\varnothing=\rho^{(0)}\to\rho^{(1)}\to
\dots\to\rho^{(l-1)}\to\rho^{(l)}=\lambda.
\end{equation*}
Moreover,
\betaql{lrpo}
c_{\lambda\mu}^{\,s\nu}(a)=\sum_{R}\sum_{k=0}^{l}
\frac{s_{\lambda}(a_{\rho^{(k)}}{\tss|\hspace{-1.5pt}|\tss} a)}{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(l)}}|)},
\end{equation}
summed over all sequences of partitions $R$
of the form
\betan
\mu=\rho^{(0)}\to\rho^{(1)}\to
\dots\to\rho^{(l-1)}\to\rho^{(l)}=\nu.
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{prop}

The last formula was given in \cite{ms:lr}
for polynomials closely related to $c_{\lambda\mu}^{\,s\nu}(a)$.
Due to \eqref{cla}, the Kostka numbers $K_{\lambda\mu}$
and the values of the irreducible
characters ${\rm{ch}\,}i^{\lambda}_{\mu}$ of the symmetric group
can be found from \eqref{kostka} and \eqref{charact}.

\betax\lambdabel{ex:valide}
If $|\lambda|=n$, then
\betan
{\rm{ch}\,}i^{\lambda}_{(1^n)}=\sum_{R}\sum_{k=1}^{n}
\frac{(|a_{\rho^{(k)}}|-|a_{\rho^{(0)}}|)^{n-1}}
{(|a_{\rho^{(k)}}|-|a_{\rho^{(1)}}|)
\ldots\wedge\ldots(|a_{\rho^{(k)}}|-|a_{\rho^{(n)}}|)}
=\sum_{R}\, 1,
\end{equation*}
which coincides with the
number of standard
$\lambda$-tableaux.
\qed
\end{example}

\subsection{Identities with dimensions of skew diagrams}

Specializing the variables by setting $a_i=-i+1$ for all $i\in\mathbb{Z}\tss$
in the expressions of Proposition~\ref{prop:ratfe},
we obtain some identities for
the Kostka numbers,
the values of the irreducible
characters and the Littlewood--Richardson coefficients
involving dimensions of skew diagrams.
Under this specialization, the double symmetric functions
become the {\it shifted symmetric functions\/} of \cite{oo:ss},
so that some of the combinatorial
results concerning the ring $\Lambda(x{\tss|\hspace{-1.5pt}|\tss} a)$
discussed above in the paper reduce to
the respective results of
\cite{oo:ss} for the ring $\Lambda^*$
of shifted symmetric functions; see also \cite{io:kc}
for an alternative description of the ring $\Lambda^*$.

For any
skew diagram $\theta$ denote by
$\dim\theta$
the number of standard $\theta$-tableaux
(i.e., row and column strict) with
entries in $\{1,2,\dots,|\theta|\}$ and set
\betan
H_{\theta}=\frac{|\theta|!}{\dim\theta}.
\end{equation*}
If $\theta$ is normal (nonskew), then
$H_{\theta}$ coincides with the product of
the hooks of $\theta$ due to the hook formula.
Under the specialization $a_i=-i+1$, for any partition
$\mu$ we have
\betan
a_{\mu}=(\mu_1,\mu_2-1,\dots).
\end{equation*}
The following formula for the values
of the double Schur functions was proved in \cite{oo:ss}:
if $\mu\subseteq \nu$, then
\betan
s_{\mu}(a_{\nu}{\tss|\hspace{-1.5pt}|\tss} a)=\frac{H_{\nu}}{H_{\nu/\mu}}.
\end{equation*}
This formula is deduced from Proposition~\ref{prop:interp}
with the use of \eqref{hoo} which takes the form
$s_{\lambda}(a_{\lambda}{\tss|\hspace{-1.5pt}|\tss} a)=H_{\lambda}$. Then \eqref{lrpo}
implies the identity for the
Littlewood--Richardson coefficients $c_{\lambda\mu}^{\,s\nu}$
which was proved in \cite{ms:lr}:
\betan
c_{\lambda\mu}^{\nu}=\sum_{\rho}(-1)^{|\nu/\rho|}
\frac{H_{\rho}}{H_{\nu/\rho}\, H_{\rho/\lambda}\, H_{\rho/\mu}},
\end{equation*}
summed over diagrams $\rho$ which contain both $\lambda$ and $\mu$,
and are contained in $\nu$.

We also have the respective consequences of
\eqref{kostka} and \eqref{charact}.
For partitions
$\mu=(1^{m_1}2^{m_2}\dots r^{m_r})$ and
$\rho=(\rho_1,\dots,\rho_l)$ set
\betan
\mathfrak{p}i_{\mu}(\rho)=\mathfrak{p}rod_{k=1}^r
\Big((1-\rho_1)^{k}+\dots+
(l-\rho_l)^{k}-1^k-\dots-l^{\,s k}\Big)^{m_k}
\end{equation*}
and
\betan
\varkappa_{\mu}(\rho)=\mathfrak{p}rod_{k=1}^r
\Big(\sum_{i_1\mathfrak{g}eqslant\dots\mathfrak{g}eqslant i_k\mathfrak{g}eqslant 1}
\rho_{i_1}(\rho_{i_2}-1)\dots (\rho_{i_k}-k+1)\Big)^{m_k}.
\end{equation*}

The following formulas are obtained by specializing
$a_i=i$ and $a_i=-i+1$, respectively, in \eqref{charact}
and \eqref{kostka}.

\begin{cor}\lambdabel{cor:kochar}
Let $\lambda$ and $\mu$ be partitions of $n$.
Then
\betan
{\rm{ch}\,}i^{\lambda}_{\mu}=\sum_{\rho\subseteq\lambda}\,
\frac{(-1)^{|\rho|}\, \mathfrak{p}i_{\mu}(\rho)}{H_{\rho}\, H_{\lambda/\rho}}
\end{equation*}
and
\betan
K_{\lambda\mu}=\sum_{\rho\subseteq\lambda}\,
\frac{(-1)^{|\lambda/\rho|}\,
\varkappa_{\mu}(\rho)}{H_{\rho}\, H_{\lambda/\rho}}.
\end{equation*}
\vskip-1.2\baselineskip
\qed
\end{cor}

\betax\lambdabel{ex:char}
Let
$\lambda=(3\, 2)$ and $\mu=(2\, 1^3)$. Then
\betan
\mathfrak{p}i_{\mu}(\rho)=-(\rho_1+\rho_2)^3\,
(\rho_1^{\,s 2}+\rho_2^{\,s 2}-2\,\rho^{}_1-4\,\rho^{}_2),
\end{equation*}
and
\betan
\begin{aligned}
H_{(3\, 2)/(1)}&=24/5,\qquad
H_{(3\, 2)/(2)}=2,\qquad
H_{(3\, 2)/(1^2)}=3,\qquad
H_{(3\, 2)/(3)}=2,\\
H_{(3\, 2)/(2\, 1)}&=1,\qquad
H_{(3\, 2)/(2\, 2)}=1,\qquad H_{(3\, 2)/(3\, 2)}=1.
\end{aligned}
\end{equation*}
Hence,
\betan
{\rm{ch}\,}i^{(3\, 2)}_{(2\, 1^3)}=
-\frac{5}{24}+\frac{32}{6}+\frac{81}{12}
-\frac{81}{3}+\frac{256}{12}-\frac{125}{24}=1.
\end{equation*}
\end{example}

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\end{document}