text:
\section{Introduction}

In 1967, Richard Kadison organised the first conference on Operator Algebras at Baton Rouge, an event that set together essentially all the experts in the subject at that time, both from Mathematics and Physics, and put the seed for much of the future developments in the subject.

Due to  a remarkable historical accident, two related breakthroughs were announced at that conference. M. Tomita distributed a (never published) preprint on modular theory \cite{Tom}, that set a profound and fundamental change of view in the subject. R. Haag presented in a talk the characterisation of equilibrium states in Quantum Statistical Mechanics at infinite volume \cite{HHW}, extending the Gibbs condition: the KMS (Kubo-Martin-Schwinger) property.

During the next year, fervid discussions on modular theory took place, in particular at Kadison's seminar on Operator Algebras at Penn. Eventually, Masamichi Takesaki provided a complete, extended, rigorous formulation of modular theory and showed that the modular group is characterised by the KMS condition \cite{Tak}. Clearly, this was the basis for a new deep interplay between Mathematics and Physics, that is still producing impressive results.

In this article, we briefly outline a path from these general basic results to some present developments where the interplay between Operator Algebras and Quantum Field Theory is playing a key role.

We refer to \cite{KR} and to \cite{Araki, Haag} for the basics on Operator Algebras and on Local Quantum Field Theory.

\section{General framework}

\subsection{Space and time}
We begin with general considerations of interdisciplinary nature about space and time, that explains why we expect Operator Algebras and modular theory to play a even more central role in Science in the future.

From a mathematical point of view, a space is a set $X$ endowed with a structure that put relations among elements (points): for example a distance, a topology, a measure. Time then allows to compare different configurations, in particular it provides a (partial) order.

Now, in Quantum Physics, space is never static (due to spontaneous creation/annihilation of particle-antiparticle pairs) so, in a sense, time should emerge out of the space structure. As said, one main property of time is that it has an order, the \textit{arrow of time}, that we can observe both macroscopically in classical physics and microscopically in quantum physics.

Classically, the arrow shows up by the second principle of thermodynamics, the entropy of an irreversible process never increases. On the other hand, in quantum mechanics, a wave function is set in a pure state after observation, an irreversible process that is unidirectional in time, so:
\[
\text{thermodynamics}\rightarrow \text{positive entropy}
\]
\[
\text{quantum mechanics}\rightarrow \text{collapse of the wave function}
\]
A known question is to relate the two arrows of time. An optimal answer would be to derive one arrow from the other. A less stringent, still interesting question is the following:
\[
\text{is there a general frame to encompass both arrows?}
\]
Of course, we keep in mind that time is a relative concept as we have learnt from Einstein's Theory of Relativity.

\subsection{Quantum Physics and Noncommutativity}
As is  known, the fundamental difference novelty of quantum physics with respect to classical physics is {\it noncommutativity}: observables are represented by selfadjoint operators on a Hilbert space ${\mathcal H}$, expectation values are given by vector states.  Heisenberg commutation relation between position and momentum  operators
\[
PQ - QP = i I
\]
shows that these two observables, in particular, do not commute.

By the spectral theorem, a selfadjoint operator is determined by its bounded continuous functional calculus, so we may restrict our attention to bounded selfadjoint operators.
A \textit{quantum system} is then given by a unital, selfadjoint algebra of bounded linear operator ${\mathcal M}$ on ${\mathcal H}$. If we fix our representation for the moment, we may assume that ${\mathcal M}$ is weakly closed, namely ${\mathcal M}$ is a von Neumann algebra.

Thus {\it observables} are selfadjoint elements $T$ of ${\mathcal M}$, {\it states} are normalised, normal, positive linear functionals ${\varphi}$,
\[
{\varphi}(T) = \text{expected value of the observable $T$ in the state ${\varphi}$\ .}
\]
We regard ${\mathcal M}$ as a \textit{noncommutative space}. More precisely, ${\mathcal M}$ is the dual of a noncommutative measure space since,
\[
\centerline{
${\mathcal M}$ abelian von Neumann algebra $\Leftrightarrow$ ${\mathcal M} = L^\infty(X,\mu)$}
\]
namely ${\mathcal M}$ is isomorphic to $L^\infty(X,\mu)$
for some measure space $(X,\mu)$.

\subsection{Thermal equilibrium states}
 A primary role in statistical mechanics is played by the equilibrium distribution, that is classically given by the Gibbs distribution.

A {\it finite quantum system} is given by a matrix algebra ${\mathfrak A}$, the Hamiltonian $H$ is a positive selfadjoint element of ${\mathfrak A}$ and the time evolution is $\tau_t = {\rm Ad}e^{itH}$. An equilibrium state ${\varphi}$ at inverse temperature $\beta$ is characterised by
the Gibbs condition that here is given by
\[
{\varphi}(X) = \frac{{\rm Tr}(e^{-\beta H}X)}{{\rm Tr}(e^{-\beta H})} \ .
\]
{\it What are the equilibrium states at infinite volume where there is no trace {\rm Tr} and no inner Hamiltonian?}

Let ${\mathfrak A}$ be a $C^*$-algebra, $\tau$ a one-parameter automorphism group of ${\mathfrak A}$.
A state ${\varphi}$ of ${\mathfrak A}$ is KMS at inverse temperature $\beta>0$ if for every $X,Y\in{\mathfrak A}$ there exists a complex function
$\ F_{XY}$, analytic on the strip $S_\beta=\{0<\Im z <\beta\}$, bounded and continuous on the closure of $S_\beta$, such that for all $t\in \mathbb R$
\textit{\begin{itemize}
\item[$(a)$] \textit{$F_{XY}(t)= {\varphi}\big(X\tau_t(Y)\big)$}
\item[$(b)$] \textit{$F_{XY}(t+i\beta)={\varphi}\big(\tau_t(Y)X\big)$}
\end{itemize}}
\vskip-0.3cm
\[
\begin{tabular}{c}
\centerline{\tiny  ${\varphi}\big(\tau_t(Y)X\big)$}\\
\hline
\centerline{$\big|${\tiny$\beta$}}
 \\
\hline
\centerline{\tiny  ${{\varphi}\big(X\tau_t(Y)\big)}$}
\end{tabular}
\]
{\it The KMS states are the equilibrium states}. The  KMS and the Gibbs condition coincide when the system is finite. (See \cite{BR}).

\subsection{Tomita-Takesaki modular theory}
Let ${\mathcal M}$ be a von Neumann algebra on a Hilbert space ${\mathcal H}$,  ${\varphi} = (\xi, \cdot\xi)$ the normal faithful state on ${\mathcal M}$ given by a cyclic and separating unit vector $\xi\in{\mathcal H}$. Then ${\mathcal M}$ linearly
embeds into ${\mathcal H}$ by evaluating an operator $T\in{\mathcal M}$ on $\xi$; the embedding identifies, as linear spaces, ${\mathcal M}$ with a dense linear subspace of ${\mathcal H}$, so the $^*$-operation on ${\mathcal M}$ gives a densely defined, anti-linear operator $S_0$ on ${\mathcal H}$
\[
\CD
{\mathcal M} @> X\mapsto X^* >\text{isometric} >{\mathcal M}
\\   @ V X\to X\xi V  V   @ V V X\to X\xi V \\
{\mathcal H} @ > S_0:X\xi\mapsto X^*\xi >\text{non isometric}> {\mathcal H}
\endCD
\]
$S_0$ is a closable operator; let
$S = \bar S_0$ be its closure. Then  $\Delta=S^*S > 0$ a positive selfadjoint operator on ${\mathcal H}$, the {\it modular operator}; $\log\Delta$ implements a canonical one-parameter automorphism group ${\sigma}$ of ${\mathcal M}$:
\[
s\in\mathbb R \mapsto {\sigma}_s^{\varphi}\in {\rm Aut}({\mathcal M})
\]
\[{\sigma}_s^{\varphi}(X) = \Delta^{is}X\Delta^{-is}
\]
the {\it modular automorphisms}.  So we have  an {\it intrinsic dynamics  associated with ${\varphi}$}. The $s$-parameter is
the {\it modular time}.

Moreover, if $S = J\Delta^{1/2}$ is the polar decomposition of $S$ we have a {\it canonical anti-isomorphism} between ${\mathcal M}$ and ${\mathcal M}'$  on ${\mathcal H}$
\[
J {\mathcal M} J = {\mathcal M}' \ .
\]
Some properties of the modular evolution are:

\medskip

$\bullet$ ${\sigma}^{\varphi}$ is a \textit{purely noncommutative} object (trivial in the commutative case).
\medskip

$\bullet$ ${\sigma}^{\varphi}$ is a KMS \textit{thermal equilibrium evolution}: If ${\varphi}(X) = \text{Tr}(\rho X)$ (type I case) then ${\sigma}_s^{\varphi}(X) = \rho^{is}X\rho^{-is}$.
The KMS inverse temperature is $\beta = -1$.
\medskip

$\bullet$ The \textit{arrow of modular time has a thermodynamical nature} as is associated with the sign of the inverse KMS temperature.
\medskip

$\bullet$  The \textit{modular time is intrinsic modulo a scaling} $s\mapsto {\sigma}^{\varphi}_{-s/\beta}$ that gives the inverse temperature $\beta$.
\medskip

$\bullet$ ${\sigma}^{\varphi}$ \textit{depends on the state ${\varphi}$ only up to inner automorphisms} (Connes' Radon-Nikodym theorem) \cite{C73}.

\subsection{Time as thermodynamical effect}
By the above considerations, \textit{
if time is the modular time, then the time arrow is associated both with positive entropy and with quantum structure.} Here is a diagram that summarises our discussion so far:
\smallskip

\centerline{quantum physics}
\centerline{$\updownarrow$}
\centerline{ KMS with positive temperature}
\centerline{
$\updownarrow$}
\centerline{modular time arrow}
\centerline{
$\updownarrow$}
\centerline{
 positive entropy}
\centerline{
$\updownarrow$}
\centerline{
thermodynamical arrow}
\bigskip

\noindent
We refer to \cite{Sew, CR, MR, BCL, Rov, Ron} for related discussions.
\subsection{Jones index}
Factors (von Neumann algebras with trivial center) are usually infinite-dimensional. For an inclusion of factors ${\mathcal N}\subset{\mathcal M}$ the Jones index $[{\mathcal M} : {\mathcal N}]$ measures the relative size of ${\mathcal N}$ in ${\mathcal M}$ of these infinite-dimensional objects. V. Jones showed \cite{Jo} a surprising result, the index values are quantised:
\textit
{\[
[{\mathcal M} : {\mathcal N}] =  4\cos^2\!\left(\frac{\pi}{n}\right)\ ,\quad n=3, 4,\dots \quad {\rm or} \quad [{\mathcal M} : {\mathcal N}]\geq 4 \ .
\]}
Jones index appears in many places both in mathematics and in physics. See \cite{K86, Hi, L89, L90, L92} for the index of an inclusion of  factors of infinite type and \cite{GL} and refs. therein for the the non-trivial center case.
\subsection{Local Quantum Field Theory}

In Quantum Field Theory, we have a quantum system with infinitely many degrees of freedom. The system is relativistic and particle creation and annihilation may occur.
\[
\text{No mathematically rigorous QFT model with interaction still exists in 3+1 dimensions. }
\]
According to R. Haag \cite{Haag}, the physical content of QFT is to be encoded in the net of local von Neumann algebras:
\[
O\ \text{spacetime regions} \mapsto \text{von Neumann algebras}\ {\mathcal A}(O),
\]
so to each region one associates the ``noncommutative functions" with support in $O$.
\medskip

A \textit{local net ${\mathcal A}$}  on the Minkowski spacetime $M$ is a map $O\subset M \mapsto  {\mathcal A}(O)\subset B({\mathcal H})$ such that associates with each spacetime region $O$ a von Neumann algebra ${\mathcal A}(O)$ on a fixed Hilbert space ${\mathcal H}$ with first principle physical requirements
(Haag-Kastler axioms, see \cite{Araki}):
\medskip

$\bullet$ \textit{\it Isotony}. $O_{1}\subset O_{2}\implies{\mathcal A}(O_{1})\subset{\mathcal A}(O_2)$
\medskip

$\bullet$ \textit{\it Locality}.  $O_1$,
$O_2$ spacelike $\implies[{\mathcal A}(O_{1}),{\mathcal A}(O_{2})]=\nolinebreak\{0\}$
\medskip

$\bullet$  \textit{\it Poincar\'e  covariance}. There exists a unitary representation $U$ of the Poincar\'e ${\mathcal P}_+^\uparrow$ group such that $U(g){\mathcal A}(O)U(g)^* = {\mathcal A}(gO)$, $g\in {\mathcal P}_+^\uparrow$
(in some context, the symmetry group could be the conformal group or other)

\medskip

$\bullet$  \textit{\it Positive energy and vacuum vector}.  The spectrum of the restriction of $U$ to the translation group is contained in the forward light cone. There is a unique (up to a phase) $U$-invariant vector $\Omega\in{\mathcal H}$, and $\Omega$ is cyclic for the algebra generated by all the local von Neumann algebras ${\mathcal A}(O)$:
\[
\text{$O\mapsto {\mathcal A}(O)$ may be viewed as a ``noncommutative chart'' }.
\]

\subsection{The Bisognano-Wichmann theorem.  Sewell's comment}
\label{BW}

By the Reeh-Schlieder theorem, the vacuum vector is cyclic and separating for the von Neumann algebra ${\mathcal A}(O)$ if both $O$ and its spacelike complement $O'$ have non-empty interior.

Let $W$ be a wedge region of the Minkowski spacetime (Rindler spacetime) say  $W$ is the region $x_1 > |t|$.
Under natural assumptions, the Bisognano-Wichmann theorem gives the vacuum modular operator and conjugation of ${\mathcal A}(W)$, in particular
\[
\Delta^{-is}_W = U(L_W(2\pi s))\ ,
\]
where $L_W$ is the boost one-parameter group (pure Lorentz transformations) in the $x_1$ direction.
\begin{figure}[h]  \begin{center}   {\footnotesize
\begin{tikzpicture}[scale=0.8]
\draw[->, very thin] (-1,0)--(4,0) ;
\draw[->,very thin] (0,-2.4)--(0,3) ;
\draw [-] (0,0) -- (2.4,-2.4);
\draw [-] (0,0) -- (2.8,2.8);
\draw [dashed, ->, thick, domain= 0:1.72] plot ({cosh({{h}})}, {sinh({{h}})});
\draw [dashed, thick, domain= 0:1.62] plot ({cosh({{h}})}, {-sinh({{h}})});
\draw (1,0) node {$\bullet$};
\draw (0.2,3.0) node {$t$};
\draw (1.5,-0.25) node{ $1/a$};
\draw (4.1,-0.25) node{ $x_1$};
\draw (4.96,1.0) node {$t = a^{-1}\sinh 2\pi s $, $x_1 =a^{-1}\cosh 2\pi s$};
\draw (5.26,-1.0) node {trajectory unif. accelerated observer $\cal O$};
\end{tikzpicture}}\end{center}
\end{figure}
Sewell \cite{Sew} proposed in this context (or more generally as in Schwarzschild spacetime case) a model independent
derivation of the   Hawking-Unruh temperature \cite{Haw} \cite{Unr}, by means of the
Bisognano-Wichmann theorem.
The evolution
\[
\left\{ \aligned & t(s) =  a^{-1}\sinh s \\
& x_1(s) =  a^{-1}\cosh s
\endaligned \right.
\]
corresponds to an observer $\cal O$ moving within $W$ with uniform
acceleration $a$, and his proper time is equal to $s/a$. $W$
is a natural horizon for this observer, since he cannot send a
signal out of $W$ and receive it back. The von Neumann algebra
${\mathcal A}(W)$ is therefore the proper (global) observable algebra for such a
mover. The
 proper time translations $\cal O$ are  given by the
one-parameter automorphism group $\alpha_{as} = {\rm Ad}U(L_W(a s))$ of ${\mathcal A}(W)$ corresponding
to the rescaled pure Lorentz transformations leaving $W$ invariant.
By the Bisognano-Wichmann theorem $\alpha_{as}$ satisfies the KMS condition at
inverse temperature
\[
\beta = \frac{2\pi}{a}
\]
with respect to  the restriction of
the vacuum state ${\varphi}$ to  ${\mathcal A}(W)$, i.e. the latter is a
thermal equilibrium state. By  Einstein's equivalence
principle, one can identify $W$ with the outside region of the black
hole and interpret the thermal outcome as a gravitational (black
hole) effect, so:
\[
\text{time is a geodesic flow, a quantum gravitational effect,}
\]
$\beta = 2\pi/a$ is indeed the Hawking-Unruh temperature. Cf. \cite{CR, MR}.

The geometric expression of the modular group associated with a bounded region in conformal QFT is given in \cite{HL}. We refer to \cite{DM} and refs. therein for a discussion of the Bisognano-Wichmann property in a general context.

\subsection{Representations. Index and statistics}
\label{Rep}

Doplicher-Haag-Roberts considered localised (DHR) \emph{representations} of a local net ${\mathcal A}$ \cite{DHR}. It turns out that a DHR representation is unitary equivalent to a localised endomorphism $\rho$ of a natural $C^*$-algebra generated by the net, one can define the statistics of $\rho$ and the {\it statistical dimension} $d(\rho)$ of $\rho$.

The {\it index-statistics relation} \cite{L89,L90} gives
$
d(\rho) = \Big[ {\mathcal A}(O) : \rho\big({\mathcal A}(O)\big)\Big]^{\frac12}
$ if $\rho$ is a DHR endomorphism localised in a causally complete region $O$. So
\[
{\text{DHR dimension} =\sqrt{\text{Jones index}}}
\]
and we have an equality between a physical index
and an analytical index that provides a bridge between two subjects. See \cite{L89, LR,L01} for a conceptual discussion on the matter.

\section{Intrinsic bounds on entropy}

\subsection{Araki's relative entropy}

Quantum Information is providing an increasingly important interplay with Quantum Field Theory, that naturally originated in the framework of quantum black hole thermodynamics (see \cite{Wald}). The first non commutative entropy notion, von Neumann's quantum entropy, was originally designed as a Quantum Mechanical version of Shannon's entropy: if a state $\psi$ is given a density matrix $\rho_\psi$
\[
\text{von Neumann entropy of $\psi$}: \ - {\rm Tr}(\rho_\psi\log\rho_\psi) \ .
\]
As said, an infinite quantum system is described by a von Neumann algebra ${\mathcal M}$ (e.g. a local von Neumann algebra ${\mathcal A}(O)$ in QFT) that is typically not of type $I$ (see \cite{L82}), no trace nor density matrix exists and von Neumann entropy is undefined. Nonetheless, the Tomita-Takesaki modular theory applies and one
may considers  \cite{Ar}
\[
\text{Araki's relative entropy}:\  S({\varphi} |\!| \psi) = -(\xi, \log\Delta_{\eta,\xi}\xi)
\]
between $\psi$ and the vacuum state ${\varphi}$, that extends Umegaki's type $I$ notion
\[
S({\varphi} |\!|\psi) = {\rm Tr}\big(\rho_{\varphi}(\log\rho_{\varphi} - \log \rho_\psi)\big)
\]
and is defined in general; here $\xi$ and $\eta$ are cyclic vector representatives of ${\varphi}$, $\psi$ and  $\Delta_{\eta,\xi}$ is the associated relative modular operator (see \cite{OP,Wit}).

$S({\varphi} |\!|\psi)$ measures how $\psi$ deviates from ${\varphi}$.
From the information theoretical viewpoint, $S({\varphi} |\!|\psi)$ is the mean value in the state ${\varphi}$ of the difference
between the information carried by the state $\psi$ and the state ${\varphi}$.
Two fundamental properties of the relative entropy are its  \textit{positivity}
\[
S({\varphi} |\!| \psi) \geq 0
\]
and \textit{monotonicity} w.r.t. a completely positive, normal unital map $\alpha$ (see below):
\[
S({\varphi}\cdot\alpha|\!|  \psi\cdot\alpha) \leq  S({\varphi} |\!|  \psi) \ .
\]
Relative entropy is one of the key concepts in information theory. \textit{We take the point of view that relative entropy is a primary concept and all other entropy notions are derived concepts}.

\subsection{Analog of the Kac-Wakimoto formula}
The root of our work relies in this formula for the incremental free energy of a black hole given in \cite{L97}
(see also \cite{KL05}).
Let $H_\rho$ be the Hamiltonian for a uniformly accelerated observer in the Minkowski spacetime with acceleration $a>0$ in a representation $\rho$ localised in the wedge for $W$ (the generator of the rescaled boost unitary group in the representation $\rho$, see Sections \ref{BW}, \ref{Rep}). We have
\textit{\begin{equation}
(\xi, e^{-s H_\rho} \xi)\big\vert_{s= \beta} = d(\rho)
\end{equation}}\label{fKW}
with $\xi$ the vacuum vector, $\beta = 2\pi/a$ the inverse Hawking-Unruh temperature and $d(\rho)^2$  is Jones' index.

The left hand side in \eqref{fKW} is a {\it generalised partition formula}, so $\log d(\rho)$ has an \textit{entropy meaning}.
(Cf. \cite{PP} for the entropy meaning of the logarithm of the index).

\subsection{Completely positive maps, quantum channels and entropy}
Let ${\mathcal N},{\mathcal M}$ be von Neumann algebras. A linear map $\alpha:{\mathcal N}\to {\mathcal M}$ is {\it completely positive} if
\[
\alpha\otimes {\rm id}_n : {\mathcal N}\otimes {\rm Mat}_n(\mathbb C) \to {\mathcal M}\otimes {\rm Mat}_n(\mathbb C)
\]
is positive for every positive integer $ n$. A completely positive map represents a {\it quantum operation}. We shall consider here only normal, unital completely positive maps.

Let ${\varphi}$ be a  faithful normal state of ${\mathcal M}$ and  $\alpha: {\mathcal N}\to{\mathcal M}$ completely positive map as above. Set
\[
{\rm H}_{\varphi} (\alpha) \equiv \sup_{({\varphi}_i)} \sum_{i}  S({\varphi} |\!|  {\varphi}_i) - S({\varphi}\cdot\alpha |\!| {\varphi}_i\cdot\alpha)
\]
with  supremum  over all finite families of positive linear functionals  ${\varphi}_i$  with $\sum_i {\varphi}_i = {\varphi}$ (one can easily define the relative entropy w.r.t. non-normalised states too. See \cite{CNT, Hi, OP} for other entropy aspects).

The \textit{conditional entropy} $H(\alpha)$ of $\alpha$ is defined by
\[
{\rm H}(\alpha) = \inf_{\varphi} {\rm H}_{\varphi}(\alpha)
\]
with infimum over all states ${\varphi}$ for $\alpha$ that are naturally represented by a cyclic vector (see \cite{LXbek}). Clearly \textit{${\rm H}(\alpha)\geq 0$} because
${\rm H}_{\varphi}(\alpha)\geq 0$ by the monotonicity of the relative entropy.

We shall say that $\alpha$ is a \textit{quantum channel} if its conditional entropy ${\rm H}(\alpha)$ is finite.

\subsection{Bimodules and completely positive maps}
Let $\alpha : {\mathcal N} \to {\mathcal M}$ be a completely positive, normal, unital map and ${\varphi}$ a faithful normal state of ${\mathcal M}$.
There exists an ${\mathcal N}-{\mathcal M}$ {\it bimodule} ${\mathcal H}_\alpha$, namely a Hilbert space ${\mathcal H}_\alpha$ equipped
with normal left/right actions $\ell_\alpha$/$r_\alpha$ of ${\mathcal N}$/${\mathcal M}$ on ${\mathcal H}_\alpha$, with a cyclic vector $\xi_\alpha\in{\mathcal H}_\alpha$ for $\ell_\alpha({\mathcal N})\vee r_\alpha({\mathcal M})$, such that
\[
(\xi_\alpha, \ell_\alpha(n)\xi_\alpha) = {\varphi}_{\rm out}(n)\, ,\quad (\xi_\alpha ,r_\alpha(m)\xi_\alpha) = {\varphi}_{\rm in}(m) \, ,
\]
with ${\varphi}_{\rm in} \equiv{\varphi}$, ${\varphi}_{\rm out} \equiv {\varphi}_{\rm in}\cdot\alpha$, see \cite{C}. The converse is true (modulo unitary equivalence) so, given ${\varphi}$ we have a one-to-one correspondence.
\[
\text{completely positive  map}\ \alpha \longleftrightarrow \text{cyclic bimodule}\ {\mathcal H}_\alpha
\]
with the above properties.
We have
\[
{H(\alpha) = \log \Ind(\alpha)}
\]
where $\Ind({\mathcal H})= \Ind({\mathcal H}_\alpha)$ is the Jones' index (more precisely, the minimal index) of $\ell_\alpha({\mathcal N})\subset r_\alpha({\mathcal M})'$, \cite{LXbek}.

\subsection{Generalisation of Stinespring dilation}
Let $\alpha: {\mathcal N}\to {\mathcal M}$ be a normal, completely positive, unital map between the von Neumann algebras ${\mathcal N}$ and ${\mathcal M}$.
We shall say that a dilation pair  $(\rho , v)$ with  $\rho: {\mathcal N}\to {\mathcal M}$ a normal homomorphism and $v\in {\mathcal M}$ an isometry such that
\[
\alpha(n) = v^*\rho(n) v\ , \quad n\in {\mathcal N}\ .
\]
$(\rho, v)$ is a  {\it minimal} dilation pair for $\alpha$ if  the left support of
$\rho({\mathcal N})v{\mathcal H}$ is equal to 1.

If ${\mathcal N}$, ${\mathcal M}$ are properly infinite (on a separable Hilbert space),  there exists a minimal dilation pair $(\rho, v)$ for $\alpha$ and is unique:
if $(\rho_1, v_1)$ is another minimal  pair, there exists a unique unitary $u\in {\mathcal M}$ such that
\[
u\rho(n) = \rho_1(n)u\, ,\quad v_1 = uv,\quad n\in{\mathcal N}\ .
\]
We have
\[
H(\alpha) = \log \Ind(\rho)\qquad \text{(minimal index)}.
\]
\subsection{Promoting modular theory to the bimodule setting}

Let ${\mathcal H}$ be an ${\mathcal N}-{\mathcal M}$-bimodule with finite Jones' index $\Ind({\mathcal H})$.
Given faithful, normal, states ${\varphi},\psi$ on ${\mathcal N}$ and ${\mathcal M}$, we define the \textit{modular operator} $\Delta_{\mathcal H}({\varphi} |\psi)$  of ${\mathcal H}$ with respect to ${\varphi},\psi$ as
\textit{\begin{equation}
\Delta_{\mathcal H}({\varphi} |\psi) \equiv d({\varphi}\cdot\ell^{-1})\big/ d(\psi\cdot r^{-1}\cdot\varepsilon) \ ;
\end{equation}}\label{modH}
here $\Delta_{\mathcal H}({\varphi} |\psi) \equiv d({\varphi}\cdot\ell^{-1})\big/ d(\psi\cdot r^{-1}\cdot\varepsilon)$ is Connes' {\it spatial derivative} \cite{C80} between the states
${\varphi}\cdot\ell^{-1}$ on $\ell_\alpha({\mathcal N})$ and $\psi\cdot r^{-1}\cdot\varepsilon$ on  $\ell_\alpha({\mathcal N})'$
and  $\varepsilon: \ell({\mathcal N})' \to r({\mathcal M})$ is the {\it minimal conditional expectation} (see \cite{Hi, L90,Y}).

\textit{$\log \Delta_{\mathcal H}({\varphi} |\psi) $} is called the \textit{modular Hamiltonian} of the bimodule ${\mathcal H}$.
The \textit{modular Hamiltonian of a quantum channel $\alpha$}
is the modular Hamiltonian of the  bimodule  ${\mathcal H}_\alpha$ is associated with $\alpha$.

\subsection{Properties of the modular Hamiltonian}
\label{Dprop}
For simplicity, let us assume here that ${\mathcal N}$, ${\mathcal M}$ are infinite factors and that our bimodules have finite index.
The modular Hamiltonian satisfies the following natural properties \cite{L18L}:
\[
\Delta^{is}_{\mathcal H}({\varphi}_1|{\varphi}_2)\ell(n)\Delta^{-is}_{\mathcal H}({\varphi}_1|{\varphi}_2) = \ell\big({\sigma}^{{\varphi}_1}_s(n)\big), \quad n\in{\mathcal N},
\]
\[
\Delta^{it}_{\mathcal H}({\varphi}_1|{\varphi}_2)r(m)\Delta^{-is}_{\mathcal H}({\varphi}_1|{\varphi}_2)= r\big({\sigma}^{{\varphi}_2}_{s}(m)\big), \quad m\in{\mathcal M},
\]
(implements the dynamics)
\[
\Delta^{is}_{\mathcal H}({\varphi}_1|{\varphi}_2) \otimes \Delta^{is}_{{\mathcal K}}({\varphi}_2|{\varphi}_3) = \Delta_{{\mathcal H}\otimes{\mathcal K}}^{is}({\varphi}_1|{\varphi}_3)
\]
(additivity of the energy)
\[
\Delta^{it}_{\bar{\mathcal H}}({\varphi}_2|{\varphi}_1) = d_{\mathcal H}^{-i2t}\, \overline{\Delta^{it}_{{\mathcal H}}({\varphi}_1|{\varphi}_2)}
\]
(charge symmetry, up to a phase).

If $T:{\mathcal H}\to{\mathcal H}'$ is a bimodule intertwiner, then
\[
T\Delta^{it}_{\mathcal H}({\varphi}_1|{\varphi}_2) = (d_{{\mathcal H}'}/d_{\mathcal H})^{it} \Delta^{it}_{{\mathcal H}'}({\varphi}_1|{\varphi}_2)T
\]
(functoriality, up to a phase).

Here above, the tensor product is Connes's {\it bimodule tensor product} \cite{C} w.r.t.  ${\varphi}_2$ and  $d_{\mathcal H} = \sqrt{\Ind({\mathcal H})}$.

\subsection{Physical Hamiltonian of a finite-index bimodule}
In order to remove the appearing of a phase in the natural properties in Sect. \ref{Dprop},
we may here easily modify the modular Hamiltonian so to fulfil the right physical requirements (additivity of energy, symmetry under charge conjugation, etc.). The operator
\[
K({\varphi}_1|{\varphi}_2)\equiv -\log \Delta_{\mathcal H}({\varphi}_1|{\varphi}_2) - \log d
\]
is the \textit{physical Hamiltonian} (at inverse temperature 1). Here $d = d_{{\mathcal H}}$.

The physical Hamiltonian at inverse temperature $\beta >0$ is then given by
\[
\beta^{-1}K({\varphi}_1|{\varphi}_2) =
-\beta^{-1}\log \Delta -\beta^{-1}\log d
\]
with $\Delta = \Delta_{\mathcal H}({\varphi}_1|{\varphi}_2)$.

There are two steps in passing from the modular Hamiltonian to the physical Hamiltonian:
\[
-\log \Delta\ \xrightarrow{\rm shifting}\  -\log \Delta - \log d\ \xrightarrow{\rm scaling}\ \beta^{-1}\big(-\log \Delta - \log d\big) \ .
\]
Notice that \textit{the shifting is \textit{intrinsic}, the scaling is to be determined by the context}.

The physical Hamiltonian then satisfies the natural properties as in Sect. \ref{Dprop}, without phase corrections.

\subsection{Modular and Physical Hamiltonians of  a quantum channel}

We now are going to compare two states of physical systems; ${\varphi}_{\rm in}$ is a suitable initial state, e.g. the
vacuum in QFT, and ${\varphi}_{\rm out}$ is a state that is reached from ${\varphi}_{\rm in}$ by some physically realisable process (quantum channel).

Let $\alpha:{\mathcal N}\to{\mathcal M}$ be a quantum channel and ${\varphi}_{\rm in}$ a faithful normal state of ${\mathcal M}$. Set ${\varphi}_{\rm out} = {\varphi}_{\rm in}\cdot\alpha$.
\[
 \Delta_\alpha \equiv  \Delta_{{\mathcal H}_\alpha}({\varphi}_{\rm in}|{\varphi}_{\rm out})\ ,
\]
\[
K_\alpha = \beta^{-1}\big(-\log \Delta_{\alpha} - \log d_{{\mathcal H}_\alpha}\big) \ .
\]
They are, respectively, the {\it modular and the physical Hamiltonian} at inverse temperature $\beta$ of the quantum channel $\alpha$, relative to the state transfer with input state ${\varphi}_{\rm in}$.

\subsection{Thermodynamical quantities}
We  define the \textit{entropy} $S\equiv S_{\alpha,{\varphi}_{\rm in}}$ of a quantum channel $\alpha$ with respect to the initial state ${\varphi}_{\rm in}$ by
\[
S = -(\hat\xi , \log \Delta_\alpha\hat\xi) \ ,
\]
where $\hat\xi$ is a vector representative of the state ${\varphi}_{\rm in} \cdot r^{-1}\cdot\varepsilon$ in ${\mathcal H}_\alpha$.

The quantity
\[
E_\alpha = (\hat\xi, K_\alpha\hat\xi)
\]
is the \textit{relative energy} with respect to the states ${\varphi}_{\rm in}$ and ${\varphi}_{\rm out}$.

The  \textit{free energy} $F_\alpha$ is now defined by the relative partition function
\[
F_\alpha =   \beta^{-1}\log(\hat\xi, e^{-\beta K_\alpha}\hat\xi) \ .
\]
As in eq. \eqref{fKW} one gets
\begin{equation}\label{f2}
\log (\hat\xi, e^{-\beta K_\alpha}\hat\xi) = \log d(\alpha) = \log \Ind(\alpha)^{1/2} = \frac12 H(\alpha) \ .
\end{equation}
Remarkably, $F_\alpha$ satisfies the \textit{thermodynamical relation} at temperature $ T = \beta^{-1}$:
\begin{equation}\label{TR}
F_\alpha = E_\alpha - TS_\alpha \ .
\end{equation}

\section{Applications}

\subsection{Landauer's bound for infinite systems}
In short, Landauer's principle states that ``information is physical" \cite{Lan}. One of its main motivations was to provide a solution to the Maxwell's demon paradox. More extensively, it can be stated as follows:

\noindent
{\it Any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information bearing degrees of freedom of the information processing apparatus or its environment} \cite{Ben}.

Let $\alpha:{\mathcal N} \to {\mathcal M}$ be a quantum channel between quantum systems ${\mathcal N}$, ${\mathcal M}$.
As a consequence of eq. \eqref{f2}, if $\alpha$ is irreversible, namely $\Ind(\alpha) > 1$, we have
\begin{equation}\label{Lb}
F_\alpha \geq \frac12 k T \log 2 \ ,
\end{equation}
where we have now inserted the Boltzmann constant $k$ (usually we use natural units), because the smallest index value greater than one is 2. (We are using the definition in \cite{LXbek}, rather than in \cite{Lan}, for the modular Hamiltonian, namely the conditional expectation is inserted in the right, rather than left, action term; this amounts to a change of sign, in particular for the free energy).

The original {\it  Landauer's lower bound} is
\[
F_\alpha \geq  k T \log 2 \ .
\]
In the finite-dimensional context, the the smallest index value greater than one is 4, so our lower bound reduces to the original Landauer's bound. Yet, our bound holds in the infinite-dimensional context, in full generality.

\subsection{ Bekenstein's bound}
The Bekenstein bound is a universal limit on the entropy that can be contained in a physical system of given size and total energy \cite{Bek}.
If
a system of total energy $E$, including rest mass, is enclosed in a sphere of radius $R$, then the entropy $S$ of the system is bounded by
\[
S\leq \lambda RE\ ,
\]
where $\lambda >0$ is a constant (the value $\lambda = 2\pi$ is often proposed).

In \cite{Ca}, H. Casini gave an interesting derivation for this bound, based on relative entropy considerations, although in the type $I$ case.
In QFT, the relative entropy $S({\varphi} |\!| \psi)$ subtracts indeed the entanglement ultraviolet divergences, common to two states ${\varphi}$ and $\psi$, and is finite for a dense set of states. A similar subtraction can be done to define a localised form of energy.

Local von Neumann algebras ${\mathcal A}(O)$ in QFT are typically of type $III$ (see \cite{L82}) and no trace does exist. By the above explained methods, we have a rigorous, general derivation of the Bekenstein upper limit for the entropy/information.

Let $\alpha: {\mathcal N} \to {\mathcal M}$ be a quantum channel between the factors ${\mathcal N}$ and ${\mathcal M}$.
As we have $F = \frac12 \beta^{-1}{\rm H}(\alpha)$ and  $H(\alpha) \geq 0$  (monotonicity of the entropy), then
\[
{F \geq 0} \quad \text{(positivity of the free energy).}
\]
So the thermodynamical relation \eqref{TR}
$
F = E - \beta^{-1} S
$
entails the general, rigorous version of the Bekenstein bound
\[
 S   \leq \beta E
\]
In particular, for a Schwarzschild black hole spacetime,
the vacuum modular group of ${\mathcal A}(O)$ is geodesic, with $O$
the black hole exterior region in the Schwarzschild-Kruskal spacetime.
The KMS temperature is the Hawking temperature
$T = 1/8\pi M = 1/ 4\pi R$
with $R= 2M$ the Schwarzschild radius, thus
\[
S \leq 4\pi R E
\]
with $S = S_{\alpha,{\varphi}}$ the entropy associated with the state transfer of the vacuum state ${\varphi}$ by a quantum channel $\alpha$, and $E$ the corresponding relative energy in the region $O$.

\section{The information in a wave packet and QNEC}
\subsection{Energy conditions in QFT}

As is known, positivity of the energy plays an important role in classical physics, general relativity in particular.
In QFT, the local energy may have negative density states \cite{EGJ}, although energy lower bounds may occur (see \cite{ FH,Wie}).

The \textit{Quantum Null Energy Condition (QNEC)} was considered in \cite{BFKLW}
(see  \cite{CF}). It is associated with a deformation $W_{\lambda}$ of a wedge $W$ in the null direction (see Figure 1 where the boundary deformation is given by the function $f_{\lambda}$, with ${\lambda}$ the deformation parameter).
By physical arguments, the QNEC reduces to the inequality
\[
S''(\lambda) \geq 0 \ .
\]
Here, for every normal faithful state $\psi$ on ${\mathcal A}(W_{\lambda})$,  $S({\lambda})$ is Araki's relative entropy $S(\psi |\!| {\varphi})$  between $\psi$ and the vacuum state ${\varphi}$ on $W_{\lambda}$ and $S''({\lambda})$ is the second derivative of $S({\lambda})$ w.r.t. ${\lambda}$.
\begin{figure}[h]
\centering
\begin{tikzpicture} \label{deform}
\draw [->] (4,0) -- (5,2);
\draw [->] (-4,0) -- (-3,2);
\draw [->] (-4,0) -- (-3,-0.5);
\draw [->] (4,0) -- (5,-0.5);
\draw [very thick, domain= 0:1] plot ({{h}}/0.5 - 1, {{{h}}- {{h}}*{{h}}*{{h}}} );
\draw[ very thick] [-] (-4,-0) -- (-1,0);
\draw[very thick] [-] (1,0) -- (4,0);
\draw[ thin] [-] (-1,0) -- (1,0);
\draw   (5.5,2) node {$u$};
 \draw (5.5,-0.5) node {$v$};
 \draw  (4.4,1.7) node {$W_{\lambda}$};
 \draw (4.2,-0.5) node {$W_{\lambda}$};
 \draw (0.06,0.1) node {$f$};
 \draw (0.34,1.2) node {$f_{\lambda}$};
 \draw [very thick, dashed, domain= 0:1] plot ({{h}}/0.5 - 1, {1.6*{{h}}- 1.6*{{h}}*{{h}}*{{h}}} );
 \draw [very thick, dashed, domain= 0:1] plot ({{h}}/0.5 - 1, {2.2*{{h}}- 2.2*{{h}}*{{h}}*{{h}}} );
\draw [-][dotted,thick] (-1,0) -- (-0.5,-0.25);
\draw [-][dotted,thick] (1,0) -- (1.5,-0.25);
\end{tikzpicture}
\caption{\footnotesize The function $f$ is the boundary of the deformed region on the null horizon. }
\end{figure}
The positivity of the second derivative of the relative entropy is surprising as it does not hold for for arbitrary systems.

In the following, we shall consider the particular, constant translation case, which is however the general case in the two-dimensional Minkowski spacetime context.

\subsubsection{Entropy of coherent states: U(1)-current model}
Let's consider the case of the $U(1)$-current $j$ on the real line and the associated local conformal net ${\mathcal A}$ of von Neumann algebras on $\mathbb R$. A localised endomorphism of ${\mathcal A}$ is associated with a real function $\ell$ in $S(\mathbb R)$, its unitary equivalence class is labeled by the charge $\int\ell$, see \cite{BMT}. We have
\[
S({\lambda})  = \pi\!\int_{{\lambda}}^{+\infty} (x-{\lambda})\ell^2(x){\rm d}x\ ,
\]
with $S({\lambda})$ vacuum relative entropy on ${\mathcal A}({\lambda}, \infty)$ of the coherent state given by the vacuum excitation by the net automorphism induced by the shift of the current $j \mapsto j + \ell$, so
\[
S'({\lambda}) = -\pi\!\int_{\lambda}^{+\infty} \ell^2(x){\rm d}x \leq 0\ ,
\]
\[
S''({\lambda}) = \pi \ell^2({\lambda}) \geq 0
\]
and we have the
\textit{positivity of $S''({\lambda})$} \cite{L19,CLR}.

We refer to \cite{LXrel} for other aspects about entropy in Conformal QFT.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale = 0.6]
\draw [->] (-0.3,0) -- (9,0);
\draw [->] (0,-0.3) -- (0,3.3);
\draw [ thick, domain=-1:1] plot ({{h}} + 2 , exp{-{{h}}*{{h}}*16});
\draw [ thick, densely dashdotted, domain=-1:1] plot ({{h}} + 2 , pi*exp{-{{h}}*{{h}}*32});
\draw [ thick, domain=-2.5 :2.5] plot ({{h}} +6, 1/4 *exp{-{{h}}*{{h}}});
\draw [ thick, dashdotted, domain=-1:1] plot ({{h}} + 6 , 1/16 *pi*exp{-{{h}}*{{h}}*2});
\draw (2.65,0.45) node { $\ell_1$};
\draw (7,0.45) node { $\ell_2$};
\end{tikzpicture}
\caption{\footnotesize Two distributions (continuous lines) for the same charge $\int\ell_1 = \int\ell_2$. The dashed lines plot the corresponding entropy density rate $S''({\lambda})$: left: high entropy, right: low entropy. }
\end{figure}

\subsection{ The entropy of a vector relative to a real linear subspace}
Let ${\mathcal H}$ be a complex Hilbert space and $H\subset {\mathcal H}$ a closed, real linear subspace.
A \textit{standard subspace} $H$ of ${\mathcal H}$ is a closed, real
linear subspace of ${\mathcal H}$ which is both \textit{cyclic} $(\overline{H +iH} ={\mathcal H})$
and \textit{separating} $(H \cap iH = \{0\})$. $H$ is {\it standard}
iff the \textit{symplectic complement} $H'=\{\xi\in H : \Im(h,k) = 0\ \forall k\in H\}$ is standard.

Let $H$ be a standard subspace; the anti-linear operator
$S : D(S) \subset {\mathcal H} \to {\mathcal H}$,
\[
S:h + ik\to h - ik, \quad h,k \in H,
\]
is involutive, closed and densely defined, indeed
$S^*_H = S_{H'}$.
Let $S = J\Delta^{1/2}$ be the polar decomposition of $S=S_H$.
Then $J$ is an anti-unitary involution, $\Delta > 0$  a non-singular, positive selfadjoint operator and
\[
\Delta^{it}H=H,\quad J H = H' \ .
\]
(one particle Tomita-Takesaki theorem), see \cite{L, LN}.
In the following, we may assume that our real linear subspace $H$ is standard and factorial, namely $1$ is not an eigenvalue of $\Delta$.

 Our analysis relies on the concept of entropy $S_k$ of a vector $k$ in a Hilbert space ${\mathcal H}$ with respect to the standard subspace $H$ of ${\mathcal H}$.
Our formula for the entropy for $S_k$ is
\[
S_k = \Im(k, P_H i \log\Delta\, k) \ .
\]
Here $P_H$ is the \textit{cutting projection}
$P_H : H + H' \to H$,
\[
P_H: h + h' \mapsto h \ .
\]
In terms of the modular data,  $P_H$ is given by
\[
P_H = \Delta^{-1/2}(\Delta^{-1/2} - \Delta^{1/2})^{-1}
+ J (\Delta^{-1/2}
- \Delta^{1/2})^{-1} \ .
\]
\subsubsection{The information carried by a classical wave}
{\it Suppose that some information is encoded and transmitted by an undulatory signal,
what is the information carried by the wave packet  in a given region at any given time?}

By a \textit{wave}, we shall here mean a solution of the Klein-Gordon equation
\[
(\square + m^2)\Phi = 0 \ ;
\]
we consider real waves  $\Phi$
with compactly supported, smooth time zero Cauchy data.

Waves' linear spacetime is equipped with an independent symplectic form given by
$\frac12\int_{t = {\rm const}}(\Phi'\Psi - \Psi'\Phi )dx$ (here the dash denotes the time derivative), which is the imaginary part of a natural complex scalar product (depending on the mass $m$); so the wave linear space is dense in a Hilbert space ${\mathcal H}$.

Waves with Cauchy data supported in the half-space $x^1 \geq 0$ form a real linear subspace $H(W)\subset{\mathcal H}$ ($W$ wedge), whose closure is standard. Then we define the entropy $S_\Phi$ of $\Phi$ as the entropy of the vector $\Phi$ with respect to the closure of $H(W)$ as above \cite{L19, CLR}.

The entropy $S_{\Phi}({\lambda})$ of  $\Phi$ w.r.t. the wedge region $W_{\lambda}$ is given by
\begin{equation}\label{Sp}
S_{\Phi}({\lambda}) = 2\pi\int_{t = {\lambda},\,  x^1\geq {\lambda}}(x^1 -{\lambda})T_{00}(t,x) dx \ ,
\end{equation}
with $ T_{00} = \frac12 \big( \sum_{\mu=0}^d (\partial_\mu \Phi)^2 + m^2\Phi^2   \big)$ the classical energy-density of $\Phi$.

In particular, the inequality
\begin{equation}\label{S2}
 S''_{\Phi}({\lambda})  \geq 0
\end{equation}
holds true.

\subsection{QNEC for coherent states and constant translations}
Let ${\mathcal A}(W)$ be the von Neumann algebra associated with the Rindler wedge by the free, neutral quantum field theory.
Given a real Klein-Gordon wave $\Phi$, consider the automorphism
\[
\beta_\Phi = {\rm Ad}V(\Phi)^*|_{{\mathcal A}(W)}
\]
and the vacuum entropy of $\beta_\Phi$ relative to the subwedge $W_{\lambda} = W + ({\lambda},{\lambda}, 0,\dots,0)$, namely $\beta_\Phi$ is the automorphism of ${\mathcal A}(W_{\lambda})$ implemented by the Weyl unitary $V(\Phi)$ associated with $\Phi$.

We have that the entropy of $\beta_\Phi$ with respect to the vacuum on $W_{\lambda}$ is given by
\[
S({\varphi}_\Phi|_{{\mathcal A}(W_{\lambda})} |\!| {\varphi}|_{{\mathcal A}(W_{\lambda})})   =  S_\Phi({\lambda}) \ ,
\]
where ${\varphi}$ is the vacuum state, ${\varphi}_\Phi = {\varphi}\cdot\beta_\Phi$ and
$S_\Phi({\lambda})$ is the entropy of the wave $\Phi$ with respect to $H(W_{\lambda})$ given in \eqref{Sp}.

In particular, the positivity of $S''_\Phi({\lambda})$ in \eqref{S2} gives
\textit{the QNEC inequality
for coherent states and constant null translations} for the free scalar quantum field case \cite{CLR}.
We refer to \cite{CLR} for more details.

We end up by summarising our construction by the following diagram:
\[
\CD
\boxed{\text {Entropy of a vector}}    @>\text{one-particle space}> \textit{classical field theory}>  \boxed{ \text{Entropy of a wave}} \\ @V\text{Fock}V
\text{space}\phantom{\bar X} V	@V\text{$2^{nd}$}V \text{quantisation}\phantom{\bar X}  V  \\  \boxed{\text{Entropy of a coherent state} }   @>\text{local algebras}> \textit{quantum field theory}>     \boxed{\text {Entropy of Klein-Gordon QFT}}\endCD
\]
So, {\it the entropy of a vector $\Phi$ with respect to a real linear subspace $H(W)$ has a double physical interpretation}: classically, it measures the information carried by a wave packet in the spacetime region $W$; from the quantum point of view, it gives the vacuum relative entropy,
on the algebra ${\mathcal A}(W)$ associated with  $W$, of the coherent state induced by $\Phi$ on the Fock space.
\section{Final comments} The above thermodynamical interpretation of time concerns a system in thermal equilibrium; this is an asymptotic situation, that may apply to our present universe, not soon after the big bang. So far, the understanding of a system out of equilibrium is definitely incomplete and mathematical methods are to be developed, in particular Operator Algebraic ones.
However, Operator Algebras have recently provided a model independent framework, with effective computational methods, for non-equilibrium thermodynamics in low dimensional conformal field theory \cite{HL}.

We finally mention that the Tomita-Takesaki modular theory also gives a powerful, surprising way to generate hidden symmetries in QFT, see \cite{Bor} and refs. therein.
\bigskip

\noindent
{\bf Acknowledgements.}
Our discussion on time was initiated to be presented at the workshop ``The Origins and Evolution of Spacetime'' in the Vatican City at the Pontifical Lateran University, on November 27--28, 2018. We thank the IRAFS Director Gianfranco Basti and vice-Director Flavia Marcacci for the kind invitation.
\smallskip

\noindent
We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The Bernoulli and Euler numbers and polynomials have been studied extensively
over the last two centuries, both for their numerous important applications
in number theory, combinatorics, numerical analysis and other areas of pure
and applied mathematics, and for their rich structures as interesting objects
in their own right. The Bernoulli numbers $B_n$, $n=0,1,2,\ldots$, can be
defined by the exponential generating function
\begin{equation}\label{1.1}
\frac{z}{e^z-1} = \sum_{n=0}^\infty B_n\frac{z^n}{n!}\qquad (|z|< 2\pi).
\end{equation}
They are rational numbers, the first few being 1, $-\frac{1}{2}$, $\frac{1}{6}$,
0, $-\frac{1}{30}$, 0, $\frac{1}{42},\ldots$, with $B_{2k+1}=0$ for $k\geq 1$.
For the most important properties see, for instance, \cite[Ch.~23]{AS} or its
successor \cite[Ch.~24]{DLMF}. Other good references are \cite{GKP},
\cite{Jo}, or \cite{No}. For a general bibliography, see \cite{DSS}.

Numerous linear and nonlinear recurrence relations for these numbers
are known, and such relations also exist for the Bernoulli {\it polynomials}
and for Euler numbers and polynomials which will be defined later.
This paper deals with {\it nonlinear\/} recurrence relations, the prototype
of which is Euler's well-known identity
\begin{equation}\label{1.2}
\sum_{j=0}^n \binom{n}{j}B_jB_{n-j} = -nB_{n-1} - (n-1)B_n\qquad (n\geq 1).
\end{equation}
This can also be seen as a convolution identity. Two different types of
convolution identities were discovered more recently, namely
\begin{equation}\label{1.3}
\sum_{j=2}^{n-2}\frac{B_jB_{n-j}}{j(n-j)}
-\sum_{j=2}^{n-2}\binom{n}{j}\frac{B_jB_{n-j}}{j(n-j)}
= 2H_n\frac{B_n}{n} \qquad (n\geq 4)
\end{equation}
by Miki \cite{Mi}, where $H_n=1+\frac{1}{2}+\dots+\frac{1}{n}$ is the $n$th
harmonic number, and
\begin{equation}\label{1.4}
(n+2)\sum_{j=2}^{n-2}B_jB_{n-j}
-2\sum_{j=2}^{n-2}\binom{n+2}{j}B_jB_{n-j} = n(n+1)B_n \qquad (n\geq 4)
\end{equation}
by Matiyasevich \cite{Ma}; see also \cite{Ag} and the references therein.
These two identities, which are remarkable in that they combine two different
types of convolutions, were later extended to Bernoulli polynomials by Gessel
\cite{Ge2} and by Pan and Sun \cite{PS}, respectively. Gessel \cite{Ge2} also
extended \eqref{1.3} to third-order convolutions, i.e., sums of products
of three Bernoulli numbers. Later Agoh \cite{Ag} found different and simpler
proofs of the polynomial analogues of \eqref{1.3} and \eqref{1.4} and proved
numerous other similar identities involving Bernoulli, Euler, and Genocchi
numbers and polynomials. Subsequently Agoh and the first author \cite{AD}
extended the polynomial analogue of \eqref{1.4} to convolution identities of
arbitrary order, and did the same for Euler polynomials. Meanwhile, following
different lines of investigation, Dunne and Schubert \cite{DS} derived
an identity that has both \eqref{1.3} and \eqref{1.4} as special cases, and
Chu \cite{Ch} obtained a large number of convolution identities, some of them
extending \eqref{1.3} and \eqref{1.4}.

It is the purpose of this paper to contribute to the recent work summarized
above and to further extend the identities \eqref{1.3} and \eqref{1.4} of
Miki and Matiyasevich. In Section~2 we state a general result concerning
second-order convolutions, and derive some consequences. In Section~3 we
introduce a symbolic notation with a related calculus, and use it to state and
prove a very general identity for Bernoulli polynomials. This is then used
in Section~4, along with some methods from probability theory, to prove a
general higher-order convolution identity which gives the main result of
Section~2 as a special case. In Section~5 we apply most of the methods from
Sections~3 and~4 to Euler polynomials and again obtain general higher-order
convolution identities. Finally, in Section~6, we state and prove several
further consequences of each of our main theorems. We conclude this paper with
some further remarks in Section~7.

\section{Identities for Bernoulli polynomials}

The {\it Bernoulli polynomials\/} can be defined by
\begin{equation}\label{2.1}
B_n(x):=\sum_{j=0}^n\binom{n}{j}B_jx^{n-j},
\end{equation}
or equivalently by the generating function
\begin{equation}\label{2.2}
\frac{ze^{xz}}{e^z-1}=\sum_{n=0}^\infty B_n(x)\frac{z^n}{n!}\qquad (|z|< 2\pi).
\end{equation}
For the first few Bernoulli polynomials, see Table~1 in Section~5.
They have the special values
\begin{equation}\label{2.3}
B_n(0) = B_n,\qquad B_n(\tfrac{1}{2})=(2^{1-n}-1)B_n,\qquad B_n(1)=(-1)^nB_n,
\end{equation}
($n=0, 1, 2,\ldots$),
where the first identity is immediate from comparing \eqref{2.2} with
\eqref{1.1}, and the other two follow from easy manipulations of the generating
function \eqref{2.2}. We also require the Pochhammer symbol (or rising
factorial) $(z)_k$, defined for $z\in\mathbb C$ and integers $k\geq 0$ by
\begin{equation}\label{2.4}
(z)_k = \frac{\Gamma(z+k)}{\Gamma(z)} = z(z+1)\dots (z+k-1),
\end{equation}
where the right-hand product is valid for $k\geq 1$.

We are now ready to state our first main result, which will be proved later.

\begin{theorem}\label{thm:Thm1}
For integers $n\geq 1$ and real numbers $a, b> 0$ we have
\begin{align}
\sum_{l=0}^{n}\binom{n}{l}\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}B_{l}(x)B_{n-l}(x)
&= \sum_{l=0}^{n}\binom{n}{l}\frac{a(b)_{l}+b(a)_{l}}{(a+b)_{l+1}}B_lB_{n-l}(x)
\label{2.5}\\
&\qquad+ \frac{ab}{(a+b+1)(a+b)}nB_{n-1}(x).\nonumber
\end{align}
\end{theorem}

The remainder of this section will be devoted to deriving a number of
consequences of this general identity. First, it is clear by \eqref{2.3} that
we get an analogous identity for Bernoulli {\it numbers} by simply deleting
the variable $x$. The most immediate special case is obtained by setting
$a=b=1$. With $(1)_n=n!$ and $(2)_n=(n+1)!$, some straightforward manipulations
involving the binomial coefficients in \eqref{2.5} lead to the following
identity, which was earlier obtained in \cite{AD}.

\begin{corollary}
For all $n\geq 1$ we have
\begin{equation}\label{2.6}
(n+2)\sum_{l=0}^n B_l(x)B_{n-l}(x)
=2\sum_{l=0}^n\binom{n+2}{l+2}B_lB_{n-l}(x)+\binom{n+2}{3}B_{n-1}(x).
\end{equation}
\end{corollary}

When $x=0$, this identity becomes trivial for odd $n$ since one of
$B_l, B_{n-l}$ will be zero except in the cases $l=1$ and $l=n-1$.
For even $n\geq 4$, however, we have the following identity.

\begin{corollary}
For all even $n\geq 2$ we have
\begin{equation}\label{2.7}
(n+2)\sum_{l=0}^n B_lB_{n-l}=2\sum_{l=0}^n\binom{n+2}{l+2}B_lB_{n-l}.
\end{equation}
\end{corollary}

This identity, although different in appearance, is equivalent to \eqref{1.4}.
For our next corollary we need the {\it shifted harmonic numbers} which for
real $a>0$ and integers $n\geq 1$ are defined by
\begin{equation}\label{2.8}
H_{a,n} := \sum_{j=0}^{n-1}\frac{1}{j+a}.
\end{equation}
Obviously, $H_{1,n}=H_n$. As we shall see, the following result can be
considered as an infinite class of generalizations of Miki's identity
\eqref{1.3}.

\begin{corollary}
For real $a>0$ and integers $n\geq 1$ we have
\begin{align}
\sum_{l=0}^{n-1}\binom{n}{l}\frac{(a)_{l}(n-l-1)}{(a)_{n}}B_{l}(x)B_{n-l}(x)
&= \sum_{l=1}^{n}\binom{n}{l}\frac{a(l-1)!+(a)_{l}}{(a)_{l+1}}B_lB_{n-l}(x)
\label{2.9}\\
&\qquad+ \frac{n}{a+1}B_{n-1}(x)+H_{a,n}B_n(x).\nonumber
\end{align}
\end{corollary}

\begin{proof}
The idea of proof is to divide both sides of \eqref{2.5} by $b$ and then take
the limit as $b\rightarrow 0$. On the left-hand side we have for
$0\leq l\leq n-1$,
\[
\frac{1}{b}\cdot\frac{(a)_l(b)_{n-l}}{(a+b)_n}
=\frac{(a)_l(b+1)\dots(b+n-l-1)}{(a+b)_n}
\rightarrow\frac{(a)_l(n-l-1)!}{(a)_n}
\]
as $b\rightarrow 0$, and on the right-hand side, for $1\leq l\leq n$,
\[
\frac{1}{b}\cdot\frac{a(b)_l+b(a)_l}{(a+b)_{l+1}}
=\frac{a(b+1)\dots(b+l-1)+(a)_l}{(a+b)_{l+1}}
\rightarrow\frac{a(l-1)!+(a)_l}{(a)_{l+1}}
\]
as $b\rightarrow 0$. To take care of the terms that were left out in the
limits above, we note that the term for $l=0$ in the right-hand sum and the
term for $l=n$ in the left-hand sum of \eqref{2.5} combine to give
\[
\binom{n}{0}\frac{a+b}{a+b}B_n(x)-\frac{(a)_n}{(a+b)_n}B_n(x)
=\frac{(a+b)_n-(a)_n}{(a+b)_n}B_n(x),
\]
and we have
\[
\lim_{b\rightarrow 0}\frac{(a+b)_n-(a)_n}{b(a+b)_n}
=\frac{1}{(a)_n}\left.\frac{d}{dx}(x)_n\right|_{x=a} = H_{a,n},
\]
where the second equation follows directly from applying the product rule
repeatedly to the right-hand side of \eqref{2.4}. Putting everything
together, we get \eqref{2.9}.
\end{proof}

As illustrations of Corollary~3 we state the cases $a=1$ and $a=2$ separately.

\begin{corollary}
For integers $n\geq 1$ we have
\begin{align}
\frac{n}{2}\sum_{l=1}^{n-1}\frac{B_l(x)}{l}\frac{B_{n-l}(x)}{n-l}
&= \sum_{l=1}^{n}\binom{n}{l}\frac{B_l}{l}B_{n-l}(x)
+\frac{n}{2}B_{n-1}(x)+H_{n-1}B_n(x),\label{2.10} \\
(n+2)\sum_{l=0}^{n-1}(l+1)B_{l}(x)&\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n+2}{l+2}(l^2+l+1)\frac{B_l}{l}B_{n-l}(x)\label{2.11}\\
&\qquad+(n+1)(n+2)\left(\frac{n}{3}B_{n-1}(x)+H_{2,n}B_n(x)\right).\nonumber
\end{align}
\end{corollary}

\begin{proof}
After some easy manipulations, \eqref{2.9} with $a=1$ gives
\begin{equation}\label{2.12}
\sum_{l=0}^{n-1}B_l(x)\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n}{l}\frac{B_l}{l}B_{n-l}(x)
+\frac{n}{2}B_{n-1}(x)+H_nB_n(x),
\end{equation}
and with $a=2$, \eqref{2.9} gives \eqref{2.11}. We now exploit the symmetry on
the left-hand side of \eqref{2.12} and rewrite the sum as
\[
\frac{B_n(x)}{n}+\frac{1}{2}\sum_{l=1}^{n-1}
\left(\frac{1}{n-l}+\frac{1}{l}\right)B_l(x)B_{n-l}(x) = \frac{B_n(x)}{n}
+\frac{n}{2}\sum_{l=1}^{n-1}\frac{B_l(x)}{l}\frac{B_{n-l}(x)}{n-l}.
\]
Finally we subtract $\frac{1}{n}B_n(x)$ from both sides of \eqref{2.12} and
note that $H_nB_n(x)$ then becomes $H_{n-1}B_n(x)$.
\end{proof}

Using a technique that involves generating functions for Stirling numbers
and N\"orlund polynomials, Gessel \cite{Ge2} obtained \eqref{2.10} as a
polynomial analogue of Miki's identity \eqref{1.3}. When $x=0$, then we also
have symmetry in the sum on the right-hand side of \eqref{2.10}; we can
therefore use again the identity $1/(n-l)+1/l=n/l(n-l)$, upon which we easily
recover Miki's identity.

Some further consequences of Theorem~1 and Corollary~3 will be derived in
the final section of this paper.

\section{Symbolic notation and general identities}

{\bf 1.} The use of symbolic notation in dealing with Bernoulli numbers and
polynomials goes back to J.~Blissard in the 1860s. Subsequently it was used
by many other authors, among them \'E.~Lucas in the 1870s and 1880s. Later it
was put on a firm theoretical foundation as part of ``the classical umbral
calculus"; see, e.g., \cite{Ge1} or \cite{RT}.

Here we propose and use a system of symbolic notation that is in some respects
similar to the classical umbral calculus, but is different in that it is
related to probability theory. Also, this system of notation is more specific
to Bernoulli numbers and polynomials and (later in this paper) Euler numbers
and polynomials.

The basis for our symbolic notation for Bernoulli numbers and polynomials are
two symbols, $\mathcal{B}$ and $\mathcal{U}$, which are complementary to each
other or, as we shall see, annihilate each other. First, we define the
{\it Bernoulli symbol\/} $\mathcal{B}$ by
\begin{equation}\label{3.1}
\mathcal{B}^{n}=B_{n}\qquad (n=0, 1,\ldots)
\end{equation}
so that, for instance, \eqref{2.1} can be rewritten as
\begin{equation}\label{3.1a}
B_n(x) = (x+\mathcal{B})^n.
\end{equation}
Furthermore, with \eqref{1.1} we have
\begin{equation}\label{3.2}
\exp\left(\mathcal{B}z\right)=\sum_{n=0}^\infty\mathcal{B}^n\frac{z^n}{n!}
=\frac{z}{e^z-1}.
\end{equation}
We also require several independent Bernoulli symbols
$\mathcal{B}_{1},\dots,\mathcal{B}_{k}$. Independence means that if we have
any two Bernoulli symbols, say $\mathcal{B}_1$ and $\mathcal{B}_2$, then
\begin{equation}\label{3.2a}
\mathcal{B}_1^{k}\mathcal{B}_2^{\ell}=B_kB_\ell.
\end{equation}
Second, the {\it uniform symbol\/} $\mathcal{U}$ is defined by
\begin{equation}\label{3.3}
f(x+\mathcal{U})=\int_0^1f(x+u)du.
\end{equation}
Here and elsewhere we assume that $f$ is an analytic function for which the
objects in question exist. From \eqref{3.3} we immediately obtain, in
analogy to \eqref{3.1},
\begin{equation}\label{3.4}
\mathcal{U}^{n}=\frac{1}{n+1}\qquad(n=0, 1,\ldots),
\end{equation}
and using this, we get
\begin{equation}\label{3.5}
\exp\left(\mathcal{U}z\right)=\sum_{n=0}^\infty\mathcal{U}^n\frac{z^n}{n!}
=\frac{e^z-1}{z}.
\end{equation}
From \eqref{3.2} and \eqref{3.5} we now deduce
\[
\exp\left(z\left(\mathcal{B}+\mathcal{U}\right)\right)
=\sum_{n=0}^\infty\left(\mathcal{B}+\mathcal{U}\right)^n\frac{z^n}{n!} = 1,
\]
which means that $\mathcal{B}$ and $\mathcal{U}$ annihilate each other,
i.e., $(\mathcal{B}+\mathcal{U})^n=0$ for all $n\neq 0$, in the sense that
\begin{equation}\label{3.6}
f(x+\mathcal{B}+\mathcal{U})=f(x),
\end{equation}
or in other words, we have the equivalence
\begin{equation}\label{3.7}
f(x)=g(x+\mathcal{U})\quad\Leftrightarrow\quad g(x)=f(x+\mathcal{B}).
\end{equation}
Finally, we note that \eqref{3.3} immediately gives, for any $u\in\mathbb R$,
\begin{equation}\label{3.8}
uf'(x+u\mathcal{U}) = f(x+u)-f(x),
\end{equation}
a difference equation that will be used repeatedly.

\medskip
{\bf 2.} It is well known that the Bernoulli polynomials are closely related
to the calculus of finite differences; see, e.g., the classic books \cite{Jo}
or \cite{No}. It is therefore not surprising that methods from difference
calculus turn out to be useful in the proofs of our main results.
Let $\Delta_u$ be the {\it forward difference operator\/} defined by
\begin{equation}\label{3.9}
\Delta_uf(x)=f(x+u)-f(x).
\end{equation}
With two (in general) distinct differences $u_1, u_2$ we compute
\begin{align*}
\Delta_{u_1}\Delta_{u_2}f(x)&=\left(f(x+u_2+u_1)-f(x+u_1)\right)
-\left(f(x+u_2)-f(x)\right) \\
&=\left(f(x+u_1+u_2)-f(x)\right)-\left(f(x+u_1)-f(x)\right)
-\left(f(x+u_2)-f(x)\right)\\
&=\Delta_{u_1+u_2}f(x)-\Delta_{u_1}f(x)-\Delta_{u_2}f(x),
\end{align*}
which gives the operator identity
\begin{equation}\label{3.10}
\Delta_{u_1+u_2}=\Delta_{u_1}\Delta_{u_2}+\Delta_{u_1}+\Delta_{u_2}.
\end{equation}
Similarly, one obtains
\begin{align}
\Delta_{u_1+u_2+u_3} &= \Delta_{u_1}\Delta_{u_2}\Delta_{u_3}
+\Delta_{u_1}\Delta_{u_2}+\Delta_{u_1}\Delta_{u_3}+\Delta_{u_2}\Delta_{u_3}\label{3.11} \\
&\quad +\Delta_{u_1}+\Delta_{u_2}+\Delta_{u_3}.\nonumber
\end{align}
To generalize these identities, we use the following notation: For a fixed
integer $k\geq 1$ and for any subset $J\subseteq\{1,\ldots,k\}$, we denote
\begin{equation}\label{3.12}
\Delta_J:=\prod_{j\in J}\Delta_{u_j},
\end{equation}
and we let $|J|$ be the cardinality of $J$. We can now state and prove the
following simple but important lemma.

\begin{lemma}
For any $k\geq 1$ and for real numbers $u_1,\ldots,u_k$ we have
\begin{equation}\label{3.13}
\Delta_{u_1+\dots+u_k}=\sum_{j=1}^k\sum_{|J|=j}\Delta_J.
\end{equation}
\end{lemma}

The case $k=1$ is trivial, and we immediately see that $k=2$ and $k=3$ give
the identities \eqref{3.10} and \eqref{3.11}, respectively.

\begin{proof}[Proof of Lemma~1]
This result can be proved by induction on $k$ in a straightforward way.
Alternatively, and more formally, we can use the shift operator
\[
f(x+u) = e^{u\partial}f(x),
\]
with the differential operator $\partial=\frac{d}{dx}$. Then we have
$\Delta_u=e^{u\partial}-1$, and
\[
\Delta_{u_1+\dots+u_k}=e^{(u_{1}+\dots+u_{k})\partial}-1
=\sum_{j=1}^k\sum_{|J|=j}\prod_{\ell\in J}\left(e^{u_{\ell}\partial}-1\right),
\]
and the result follows.
\end{proof}

{\bf 3.} We now apply results from the first two parts of this section to
obtain a general identity for Bernoulli symbols, and thus for Bernoulli numbers
and polynomials. In what follows, we assume that for a fixed integer $k\geq 1$,
$u_1,\ldots,u_k$ are real parameters. To simplify notations, we write, for a
subset $J\subseteq\{1,\ldots,k\}$,
\begin{equation}\label{3.14}
u_{J} := \prod_{j\in J}u_{j},\qquad
\left(u\mathcal{B}\right)_{J} := \sum_{j\in J}u_{j}\mathcal{B}_{j},\qquad
\overline{J}=\{1,\dots,k\}\setminus J.
\end{equation}
The following is, in fact, a restatement of an intermediate result in \cite{AD}.

\begin{lemma}
Let $u_1+\dots+u_k=1$. Then we have
\begin{equation}\label{3.15}
\frac{1}{n!}\left(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^n
=\sum_{j=1}^k\sum_{|J|=j}\frac{u_J}{(n+1-j)!}
\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right)^{n-j+1},
\end{equation}
where $\mathcal{B}_0,\dots,\mathcal{B}_k$ are independent Bernoulli symbols.
\end{lemma}

\begin{proof}
We apply the operator identity \eqref{3.13} to the function
\[
f(x):=\frac{1}{(n+1)!}
\left(x+\mathcal{B}_0+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^{n+1}.
\]
Then the left-hand side of \eqref{3.13} gives, with \eqref{3.8},
\begin{align}
\Delta_{u_1+\dots+u_k}f(x) &= \Delta_1f(x)=f(x+1)-f(x)\label{3.16}\\
&= f'(x+\mathcal{U})
=\frac{1}{n!}\left(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k\right)^n,\nonumber
\end{align}
where in the last step we used \eqref{3.6}, i.e., $\mathcal{B}_0$ is
annihilated by $\mathcal{U}$. Similarly, we have for any $i=1,\ldots, k$, again
using \eqref{3.8},
\begin{align}
\Delta_{u_i}f(x) &= f(x+u_i)-f(x) = u_if'(x+u_i\mathcal{U}) \label{3.17}\\
&= u_i\frac{1}{n!}\left(x+\mathcal{B}_0
+(u\mathcal{B})_{\{1,\ldots,k\}\setminus\{i\}}\right)^n,\nonumber
\end{align}
having used the fact that the uniform symbol $\mathcal{U}$ annihilated the
Bernoulli symbol $\mathcal{B}_i$; note that the coefficients $u_i$ have to
match for the annihilation (i.e., identity \eqref{3.6}) to apply. Using the
definition \eqref{3.12} and successively applying \eqref{3.17}, we get
\[
\Delta_Jf(x)=\frac{u_J}{(n-j+1)!}
\left(x+\mathcal{B}_{0}+(u\mathcal{B})_{\overline{J}}\right)^{n+1-j}.
\]
Finally, applying \eqref{3.13} to this and to \eqref{3.16}, we immediately
get \eqref{3.15}.
\end{proof}

While the case $k=1$ is trivial, for $k=2$ and $k=3$ we get the following
identities.

\medskip
\noindent
{\bf Examples.} For $u_1+u_2=1$, we have
\begin{align*}
\left(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2\right)^n
&= u_1\left(x+\mathcal{B}_0+u_2\mathcal{B}_2\right)^n
+u_2\left(x+\mathcal{B}_0+u_1\mathcal{B}_1\right)^n\\
&\quad +u_1u_2n\left(x+\mathcal{B}_0\right)^{n-1},
\end{align*}
and for $u_1+u_2+u_2=1$,
\begin{align*}
&\left(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2+u_3\mathcal{B}_3\right)^n
= \left[u_1\left(x+\mathcal{B}_0+u_2\mathcal{B}_2+u_3\mathcal{B}_3\right)^n+o.t.\right]\\
&\qquad+\left[nu_1u_2\left(x+\mathcal{B}_0+u_3\mathcal{B}_3\right)^{n-1}+o.t.\right]
+n\left(n-1\right)u_1u_2u_3\left(x+\mathcal{B}_0\right)^{n-2},
\end{align*}
where ``o.t." in each of the first and second rows stands for the ``other terms"
obtained by cyclically permuting the subscripts $\{1,2,3\}$.

\medskip
\noindent
{\bf Remarks.} (1) The left-hand side of \eqref{3.15}, and in fact also the
terms on the right-hand side, could be written as Bernoulli polynomials of
higher order, as defined in identity (30) in \cite[p.~39]{EMOT}. We will not
pursue this further.

(2) It is clear from the proof of Lemma~2 that, more generally,
for any analytic function $f$ and $u_1+\dots+u_k=1$ we have
\[
f(x+u_1\mathcal{B}_1+\dots+u_k\mathcal{B}_k)
=\sum_{j=1}^n\sum_{|J|=j}u_J
f^{(j-1)}\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right),
\]
and in particular, for $k=2$ and $u_1+u_2=1$,
\begin{align*}
f(x+u_1\mathcal{B}_1+u_2\mathcal{B}_2)
&=u_2f(x+\mathcal{B}_0+u_1\mathcal{B}_1)+u_1f(x+\mathcal{B}_0+u_2\mathcal{B}_2)\\
&\quad +u_1u_2f'(x+\mathcal{B}_0).
\end{align*}

The main results of this paper are based on Lemma~2 and an analogue for Euler
polynomials, and will be obtained by considering the expectation when
$u_1,\ldots,u_k$ are taken to be certain random variables.

\section{Generalization and proof of Theorem~1}

{\bf 1.} In this section we prove a higher-order analogue of Theorem~1, of
which the latter is an immediate consequence. The proof uses some probabilistic
methods which will be summarized in a brief subsection.

\begin{theorem}
For integers $k\geq 2$ and $n\geq 0$ and for positive real parameters
$a_1,\ldots,a_k$ we have
\begin{align}
&\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}B_{l_1}(x)\dots B_{l_k}(x)
=\sum_{j=1}^k\sum_{|J|=j}\frac{a_J n!}{(n+1-j)!}\label{4.0} \\
&\qquad\times\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}}
B_{l_0}(x)B_{l_1}\dots B_{l_{k-j}}.\nonumber
\end{align}
\end{theorem}

When $k=2$, we immediately get Theorem~1. For $k=3$ and $a_1=a_2=a_3=1$ we get,
after some easy transformations and renaming the summation indices,
\begin{gather}
(n+3)\sum_{i+j+l=n}B_i(x)B_j(x)B_l(x)
= 3\sum_{i+j+l=n}\binom{n+3}{i}B_i(x)B_jB_l\label{4.0a}\\
\qquad+3\sum_{i+j=n-1}\binom{n+3}{i}B_i(x)B_j
+\binom{n+3}{5}B_{n-2}(x),\nonumber
\end{gather}
valid for $n\geq 3$; this is Corollary~1 in \cite{AD}. Other special cases
with $k=3$ will be considered
later, in Section~6. For arbitrary $k\geq 2$, with $a_1=\dots=a_k=1$, we
recover Theorem~1 in \cite{AD}, which for $x=0$ gives a $k$th order analogue
of Matiyasevich's identity \eqref{1.4}, namely
\[
\sum_{l_1+\dots+l_k=n}B_{l_1}\dots B_{l_k}
=\frac{1}{n+k}\sum_{j=1}^k\binom{k}{j}
\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+k}{l_0}
B_{l_0}B_{l_1}\dots B_{l_{k-j}}.
\]

{\bf 2.} We now summarize some facts from probability theory that will be
used in the proofs that follow. For the basics we refer the reader to any
introductory text in probability theory, e.g., \cite{Du} or \cite{Wa}.
For the interplay between probability theory and umbral calculus, see
\cite{SW}.

We assume that $X$ is a continuous random variable with probability density
function $f_X(x)$, i.e.,
\begin{equation}\label{4.1}
{\rm Pr}(X\leq x) = \int_{-\infty}^x f_X(y)dy.
\end{equation}
Given a measurable function $g:{\mathbb R}\rightarrow {\mathbb R}$ such that
the image random variable $g(X)$ is absolutely integrable, its expectation
can be expressed as
\begin{equation}\label{4.2}
{\mathbb E}g(X) = \int_{-\infty}^\infty g(y)f_X(y)dy.
\end{equation}
The main tool in this section is the use of random variables with a gamma
distribution of ``scale parameter" 1. We write such a random variable as
$X \sim\Gamma_a$, with ``shape parameter" $a>0$, defined by the density function
\begin{equation}\label{4.3}
f_X(x;a) = \begin{cases}
\tfrac{1}{\Gamma(a)}x^{a-1}e^{-x} &\hbox{for}\; x\geq 0,\\
0 &\hbox{otherwise}.
\end{cases}
\end{equation}
Then from the definition of the gamma function,
\[
\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}dx,
\]
and with \eqref{4.2} and \eqref{4.3} we immediately get, for an integer
$n\geq 1$,
\begin{equation}\label{4.4}
{\mathbb E}(\Gamma_a^n) = \int_0^\infty y^n\tfrac{1}{\Gamma(a)}y^{a-1}e^{-y}dy
=\frac{\Gamma(a+n)}{\Gamma(a)} = (a)_n.
\end{equation}
An essential property of the gamma distribution is additivity, i.e., if
$\Gamma_{a_1},\ldots,\Gamma_{a_k}$ are independent gamma distributed random
variables, then
\begin{equation}\label{4.5}
\Gamma_{a_1}+\dots+\Gamma_{a_k} \sim \Gamma_{a_1+\dots+a_k},
\end{equation}
where the symbol $\sim$ indicates that the random variables on both sides have
the same distribution. The relation \eqref{4.5} follows from the fact that the
density probability function for the sum of two independent random variables
is the convolution of the individual ones; see, e.g., \cite[p.~107]{Wa}.

The next important tool is the choice of random coefficients $u_1,\ldots, u_k$
such that $(u_1,\ldots, u_k)$ follows a Dirichlet distribution with parameters
$(a_1,\ldots, a_k)$. This is equivalent to choosing $k$ independent gamma
random variables $\Gamma_{a_i}$, each having shape parameter $a_i$, and to
define
\begin{equation}\label{4.6}
u_i=\frac{\Gamma_{a_i}}{\Gamma_{a_1}+\dots+\Gamma_{a_k}},\qquad 1\leq i\leq k;
\end{equation}
note that $u_1+\dots+u_k=1$.
For Dirichlet distributions in general, see, e.g., \cite[p.~231]{JK}. We now
need an important property of gamma random variables, namely that
$\Gamma_a+\Gamma_b$ and $\Gamma_a/(\Gamma_a+\Gamma_b)$ are independent when
$\Gamma_a$ and $\Gamma_b$ are. In fact, this characterizes gamma random
variables; see \cite{Lu}. This is easily extended to the
statement that
\begin{equation}\label{4.7}
\Gamma_{a_1}+\dots+\Gamma_{a_k}\quad\hbox{and}\quad
\frac{\Gamma_{a_i}}{\Gamma_{a_1}+\dots+\Gamma_{a_k}},\qquad 1\leq i\leq k,
\end{equation}
are independent. The importance of this lies in the fact that
${\mathbb E}(XY)={\mathbb E}(X){\mathbb E}(Y)$ for independent random
variables $X$ and $Y$.

Combining all of the above, we first note that for any positive integers
$l_1,\ldots, l_k$ we have by \eqref{4.6},
\begin{align}
{\mathbb E}[(\Gamma_{a_1}+\dots+\Gamma_{a_k})^{l_1+\dots+l_k}
(u_1^{l_1}\dots u_k^{l_k})]
&={\mathbb E}(\Gamma_{a_1}^{l_1}\dots\Gamma_{a_k}^{l_k})\label{4.8}\\
&={\mathbb E}(\Gamma_{a_1}^{l_1})\dots
{\mathbb E}(\Gamma_{a_k}^{l_k}).\nonumber
\end{align}
On the other hand, using the independence of the terms in \eqref{4.7}, we see
that the left-hand side of \eqref{4.8} is equal to
\begin{equation}\label{4.9}
{\mathbb E}(\Gamma_{a_1+\dots+a_k}^{l_1+\dots+l_k})
{\mathbb E}(u_1^{l_1}\dots u_k^{l_k}),
\end{equation}
having also used \eqref{4.5}. Finally, applying \eqref{4.4} to the right-hand
side of \eqref{4.8} and to \eqref{4.9}, we get
\begin{equation}\label{4.10}
{\mathbb E}(u_1^{l_1}\dots u_k^{l_k})
=\frac{(a_1)_{l_1}\dots (a_k)_{l_k}}{(a_1+\dots+a_k)_{l_1+\dots+l_k}}.
\end{equation}
This identity will be used repeatedly in what follows.

\medskip
{\bf 3.} We are now ready to prove Theorem~2; as we shall see, much of the
work was already done in obtaining the identities \eqref{3.15} and \eqref{4.10}.

\begin{proof}[Proof of Theorem~2]
We choose $u_1,\ldots, u_k$ as in \eqref{4.6}. Since $u_1+\dots+u_k=1$, we can
rewrite the $n$th power term on the left-hand side of \eqref{3.15} as follows,
and then apply a multinomial expansion, using \eqref{3.1a}:
\begin{align}
&\left(u_1(x+\mathcal{B}_1)+\dots+u_k(x+\mathcal{B}_k)\right)^n\label{4.11}\\
&\qquad\qquad=\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
u_1^{l_1}\dots u_k^{l_k}B_{l_1}(x)\dots B_{l_k}(x).\nonumber
\end{align}
Similarly, we use multinomial expansions for the powers on the right of
\eqref{3.15}, this time combining the terms $x+\mathcal{B}_0$ for the sake of
applying \eqref{3.1a}:
\begin{align}
&\left(x+\mathcal{B}_0+(u\mathcal{B})_{\overline{J}}\right)^{n-j+1}\label{4.12}\\
&\qquad=\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
B_{l_0}(x)\left(u_{i_{j+1}}\mathcal{B}_{i_{j+1}}\right)^{l_1}\dots
\left(u_{i_k}\mathcal{B}_{i_k}\right)^{l_{k-j}}\nonumber\\
&\qquad=\sum_{\substack{l_0+l_1+\dots+l_{k-j}\\=n+1-j}}\binom{n+1-j}{l_0,l_1,\ldots,l_{k-j}}
u_{i_{j+1}}^{l_1}\dots u_{i_k}^{l_{k-j}}B_{l_0}(x)B_{l_1}\dots B_{l_{k-j}},\nonumber
\end{align}
where we have also used \eqref{3.2a}. All that remains to be done now is to
compute the expectation on both sides of \eqref{3.15}, which mainly involves
applying \eqref{4.10} to the right-hand sides of \eqref{4.11} and \eqref{4.12}.
In particular, keeping the first notation in \eqref{3.14} in mind, we have
\begin{align*}
{\mathbb E}(u_Ju_{i_{j+1}}^{l_1}\dots u_{i_k}^{l_{k-j}})
&=\frac{(a_{i_1})_1\dots(a_{i_j})_1(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}
{(a_1+\dots+a_k)_{n+1-l_0}} \\
&=a_J\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}},
\end{align*}
where we have used the fact that $(a)_1=a$ and, in the denominator, that
$1+\dots +l_1+\dots+l_{k-j}=j+(n-j+1)-l_0=n+1-l_0$. This completes the proof.
\end{proof}

\section{Euler numbers and polynomials}

The Euler numbers and polynomials are often considered in parallel with their
Bernoulli analogues. Indeed, they are similar in various respects, including
their importance in the classical calculus of finite differences (see, e.g.,
\cite{Jo} or \cite{No}). In this section we follow the outlines of the previous
sections to derive analogous results for Euler polynomials and, to a lesser
extent, Euler and Genocchi numbers.

\medskip
{\bf 1.} The {\it Euler numbers\/} $E_n$, $n=0, 1, 2,\ldots$,
can be defined by
\begin{equation}\label{5.1}
\frac{2}{e^z+e^{-z}} = \sum_{n=0}^\infty E_n\frac{z^n}{n!}\qquad
(|z|< \tfrac{\pi}{2}).
\end{equation}
The Euler numbers are all integers with $E_n=0$ when $n$ is odd; the first
few values are listed in Table~1. The {\it Euler polynomials\/} can be
defined by
\begin{equation}\label{5.2}
E_n(x):=\sum_{j=0}^n\binom{n}{j}\frac{E_j}{2^j}(x-\tfrac{1}{2})^{n-j},
\end{equation}
or equivalently by the generating function
\begin{equation}\label{5.3}
\frac{2e^{xz}}{e^z+1}=\sum_{n=0}^\infty E_n(x)\frac{z^n}{n!}\qquad (|z|< \pi).
\end{equation}
A key consequence of \eqref{5.3} is the functional equation
\begin{equation}\label{5.4}
E_n(x)+E_n(x+1)=2x^n,\qquad n=0,1,2,\ldots,
\end{equation}
which gives rise to numerous applications.
One important difference to the Bernoulli case is the fact that $E_n(0)$ is
{\it not\/} the $n$th Euler number. The {\it Genocchi numbers\/} $G_n$, are
often used instead; they are closely related to the Bernoulli numbers via
\begin{equation}\label{5.5}
G_n := 2(1-2^n)B_n\qquad (n=0, 1, 2,\ldots).
\end{equation}
These numbers are all integers; the first few values are also listed
in Table~1.

\bigskip
\begin{center}
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|r||r|r|r|l|l|}
\hline
$n$ & $B_n$ & $E_n$ & $G_n$ & $B_n(x)$ & $E_n(x)$\\
\hline
0 & 1 & 1 & 0 & 1 & 1 \\
1 & $-1/2$ & 0 & 1 & $x-\tfrac{1}{2}$ & $x-\tfrac{1}{2}$ \\
2 & $1/6$ & $-1$ & $-1$ & $x^2-x+\tfrac{1}{6}$ & $x^2-x$ \\
3 & 0 & 0 & 0 & $x^3-\tfrac{3}{2}x^2+\tfrac{1}{2}x$ & $x^3-\tfrac{3}{2}x^2+\tfrac{1}{4}$ \\
4 & $-1/30$ & 5 & 1 & $x^4-2x^3+x^2-\tfrac{1}{30}$ &
$x^4-2x^3+x$ \\
5 & 0 & 0 & 0 & $x^5-\tfrac{5}{2}x^4+\tfrac{5}{3}x^3-\tfrac{1}{6}x$ &
$x^5-\tfrac{5}{2}x^4+\tfrac{5}{2}x^2-\tfrac{1}{2}$ \\
6 & $1/42$ & $-61$ & $-3$ & $x^6-3x^5+\tfrac{5}{2}x^4-\tfrac{1}{2}x^2+\tfrac{1}{42}$ &
$x^6-3x^5+5x^3-3x$ \\
\hline
\end{tabular}}

\medskip
{\bf Table~1}: $B_n, E_n, G_n, B_n(x)$ and $E_n(x)$ for $0\leq k\leq 6$.
\end{center}
\bigskip

By elementary manipulations of the relevant generating functions, we get
\begin{equation}\label{5.6}
E_n(0)=\frac{1}{n+1}G_{n+1},\qquad E_n(\tfrac{1}{2})=2^{-n}E_n \qquad
(n=0, 1, 2,\ldots).
\end{equation}
The Euler polynomial analogue of Theorem~1 can now be stated as follows.

\begin{theorem}
For integers $n\geq 1$ and real numbers $a, b> 0$ we have
\begin{align}
\sum_{l=0}^{n}\binom{n}{l}&\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}E_{l}(x)E_{n-l}(x)
= \frac{4}{n+1}B_{n+1}(x) \label{5.6a}\\
&-\frac{2}{n+1}\sum_{l=0}^{n+1}\binom{n+1}{l}
\frac{(a)_{l}+(b)_{l}}{(a+b)_l}E_l(0)B_{n+1-l}(x)\nonumber
\end{align}
\end{theorem}
As in the case of Theorem~1, this result follows from a higher-order
convolution identity that will be proved later. As a special case of
\eqref{5.6a}, for $a=b=1$, we get the following Euler polynomial analogue
of Matiyasevich's identity:
\[
(n+2)\sum_{l=0}^{n}\binom{n}{l}E_{l}(x)E_{n-l}(x)
=4(n+2)B_{n+1}(x)-4\sum_{l=0}^{n+1}\binom{n+2}{l}B_l(x)E_{n+1-l}(0).
\]
This identity was earlier obtained as Corollary~2 in \cite{AD}.

\medskip
{\bf 2.} As we develop a formalism parallel to that involving the Bernoulli
symbol, we note that the analogue of $B_n$ is $E_n(0)$. Thus, we define the
{\it Euler symbol} $\mathcal E$ by
\begin{equation}\label{5.7}
{\mathcal E}^n = E_n(0),\qquad n=0,1,2,\ldots,
\end{equation}
and elementary manipulation of the generating function \eqref{5.3} gives
\begin{equation}\label{5.8}
E_n(x) = (x+{\mathcal E})^n,\qquad n=0,1,2,\ldots.
\end{equation}
The analogue to the uniform symbol $\mathcal U$ defined in Section~3 is the
uniform discrete symbol $\mathcal V$ with generating function
\begin{equation}\label{5.9}
e^{z{\mathcal V}} = \tfrac{1}{2}+\tfrac{1}{2}e^z,
\end{equation}
or equivalently defined by
\[
f(x+\mathcal{V}) = \frac{f(x)+f(x+1)}{2}
\]
for an analytic function $f$; this is a discrete analogue of \eqref{3.3}.
With a change of variable we have for any real $u$,
\begin{equation}\label{5.10}
f(x+u{\mathcal V}) = \frac{1}{2}f(x)+\frac{1}{2}f(x+u),
\end{equation}
which is analogous to \eqref{3.8}, and which will be just as useful. Next,
by multiplying \eqref{5.3}, setting $x=0$, with \eqref{5.9}, we see that in
analogy with \eqref{3.6} and \eqref{3.7} we have ${\mathcal E}+{\mathcal V}=0$
in the sense that
\begin{equation}\label{5.11}
f(x+{\mathcal E}+{\mathcal V}) = f(x),
\end{equation}
or in other words,
\begin{equation}\label{5.12}
f(x)=g(x+\mathcal{V})\quad\Leftrightarrow\quad g(x)=f(x+\mathcal{E}).
\end{equation}

{\bf 3.} The functional equations \eqref{5.4} and \eqref{5.10} give rise to
the definition of the {\it discrete forward difference operator} $\delta_u$
defined by
\begin{equation}\label{5.13}
\delta_uf(x) = \frac{f(x)+f(x+u)}{2}.
\end{equation}
Thus, in particular, we have $\delta_1E_n(x)=x^n$ and by \eqref{5.10},
\begin{equation}\label{5.14}
\delta_uf(x) = f(x+u{\mathcal V}).
\end{equation}
In analogy to \eqref{3.10} we now
compute
\begin{align*}
2\delta_{u_1}\delta_{u_2}f(x)
&=\frac{f(x)+f(x+u_2)}{2}+\frac{f(x+u_1)+f(x+u_2+u_1)}{2} \\
&=\frac{f(x+)+f(x+u_1+u_2)}{2}+\frac{f(x)+f(x+u_1)}{2}\\
&\qquad+\frac{f(x)+f(x+u_2)}{2}-f(x),
\end{align*}
which gives the operator identity
\begin{equation}\label{5.15}
\delta_{u_1+u_2}=2\delta_{u_1}\delta_{u_2}-\delta_{u_1}-\delta_{u_2}+1.
\end{equation}
Similarly, one obtains
\begin{align}
\delta_{u_1+u_2+u_3} &= 4\delta_{u_1}\delta_{u_2}\delta_{u_3}
-2\delta_{u_1}\delta_{u_2}-2\delta_{u_1}\delta_{u_3}-2\delta_{u_2}\delta_{u_3}\label{5.16} \\
&\qquad +\delta_{u_1}+\delta_{u_2}+\delta_{u_3}.\nonumber
\end{align}
In general, using the notation $\delta_J$, with the same meaning as in
\eqref{3.12}, where again $J\subseteq\{1,\ldots,k\}$, we have the following
result.

\begin{lemma}
For even $k\geq 2$ we have
\begin{equation}\label{5.17}
\delta_{u_1+\dots+u_k} = 1-\sum_{j=1}^k\sum_{|J|=j}(-2)^{j-1}\delta_J,
\end{equation}
and for odd $k\geq 1$,
\begin{equation}\label{5.18}
\delta_{u_1+\dots+u_k} = \sum_{j=1}^k\sum_{|J|=j}(-2)^{j-1}\delta_J.
\end{equation}
\end{lemma}

These identities can be proved by straightforward induction, with \eqref{5.15}
as induction beginning. Using notation from \eqref{3.14}, we now
obtain the following result.

\begin{lemma}
Let $u_1+\dots+u_k=1$. Then for even $k\geq 2$ we have
\begin{equation}\label{5.19}
(n+1)\left(x+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n
=\sum_{j=1}^k(-2)^j\sum_{|J|=j}
\left(x+\mathcal{B}+(u\mathcal{E})_{\overline{J}}\right)^{n+1},
\end{equation}
and for odd $k\geq 1$,
\begin{equation}\label{5.20}
\left(x+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n
=\sum_{j=1}^k(-2)^{j-1}\sum_{|J|=j}
\left(x+\mathcal{E}_0+(u\mathcal{E})_{\overline{J}}\right)^n,
\end{equation}
where $\mathcal{E}_0,\dots,\mathcal{E}_k$ are independent Euler symbols.
\end{lemma}

\begin{proof}
Comparing \eqref{5.13} with \eqref{3.9}, we get the operator identity
$\Delta_u = 2\delta_u-2$,
and thus for even $k\geq 2$ we have with \eqref{5.17},
\begin{equation}\label{5.22}
\Delta_{u_1+\dots+u_k} = \sum_{j=1}^k(-2)^j\sum_{|J|=j}\delta_J.
\end{equation}
Since $u_1+\dots u_k=1$, we have by \eqref{3.16},
\begin{equation}\label{5.23}
\Delta_{u_1+\dots+u_k}f(x) = f'(x+\mathcal{U}).
\end{equation}
Now let
\[
f(x):=\left(x+\mathcal{B}+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^{n+1},
\]
and apply \eqref{5.22} to this function. On the left-hand side, using
\eqref{5.23}, the symbols $\mathcal{U}$ and $\mathcal{B}$ cancel each other,
and we get the left-hand side of \eqref{5.19}. To obtain the right-hand side,
we first note that for any $i=1,\ldots,k$ we have by \eqref{5.14},
\begin{equation}\label{5.24}
\delta_{u_i}f(x)
=\left(x+\mathcal{B}+(u\mathcal{E})_{\{1,\ldots,k\}\setminus\{i\}}\right)^{n+1},
\end{equation}
having used the notation in \eqref{3.14} and the fact that the symbols
$\mathcal{V}$ and $\mathcal{E}_i$ cancel each other. As in
\eqref{3.17}, the coefficients $u_i$ have to match for this cancellation to
apply. Successively applying \eqref{5.24} and using the notation \eqref{3.12},
we get
\[
\delta_Jf(x) = \left(x+\mathcal{B}+(u\mathcal{E})_{\overline{J}}\right)^{n+1}.
\]
This, combined with \eqref{5.22}, completes the proof of \eqref{5.19}.

The proof of \eqref{5.20} is very similar: Instead of \eqref{5.22} use
\eqref{5.18} and apply it to
\[
f(x):=\left(x+\mathcal{E}_0+u_1\mathcal{E}_1+\dots+u_k\mathcal{E}_k\right)^n.
\]
While the right-hand side is evaluated as before, for the left-hand side we
use \eqref{5.14} with $u=1$.
\end{proof}

{\bf 4.} We are now ready to state and prove the main result of this section.

\begin{theorem}
Let $n\geq 0$ and $k\geq $ be integers and $a_1,\ldots,a_k$ positive
parameters. Then for even $k\geq 2$ we have
\begin{gather}
\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}E_{l_1}(x)\dots E_{l_k}(x)
=\sum_{j=1}^k\frac{(-2)^j}{n+1}\label{5.25} \\
\quad\times\sum_{|J|=j}\sum_{\substack{l_0+l_1+\dots\\+l_{k-j}=n+1}}\binom{n+1}{l_0,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n+1-l_0}}
B_{l_0}(x)E_{l_1}(0)\dots E_{l_{k-j}}(0),\nonumber
\end{gather}
and for odd $k\geq 1$,
\begin{gather}
\sum_{l_1+\dots+l_k=n}\binom{n}{l_1,\ldots,l_k}
\frac{(a_1)_{l_1}\dots(a_k)_{l_k}}{(a_1+\dots+a_k)_n}E_{l_1}(x)\dots E_{l_k}(x)
=\sum_{j=1}^k(-2)^{j-1}\label{5.26} \\
\quad\times\sum_{|J|=j}\sum_{\substack{l_0+l_1+\dots\\+l_{k-j}=n}}\binom{n}{l_0,\ldots,l_{k-j}}
\frac{(a_{i_{j+1}})_{l_1}\dots(a_{i_k})_{l_{k-j}}}{(a_1+\dots+a_k)_{n-l_0}}
E_{l_0}(x)E_{l_1}(0)\dots E_{l_{k-j}}(0).\nonumber
\end{gather}
\end{theorem}

For $k=2$, the identity \eqref{5.25} reduces to Theorem~3.
In the special case $a_1=\dots=a_k=1$, the identities \eqref{5.25} and
\eqref{5.26} reduce to Theorems~2 and~3, respectively, in \cite{AD}. Other
special cases can be found in Section~6.

\begin{proof}[Proof of Theorem~4]
The proof is almost identical to that of Theorem~2: We expand the powers on
both sides of \eqref{5.19} and \eqref{5.20} using the multinomial theorem,
and then compute the expectation on both sides by way of \eqref{4.10}, having
chosen $u_1,\ldots,u_k$ as in \eqref{4.6}.

The left-hand sides of \eqref{5.25} and \eqref{5.26} are obtained just as in
the expansion \eqref{4.11}, with Euler instead of Bernoulli symbols and
polynomials, and having used \eqref{5.8} in place of \eqref{3.1a}. The
right-hand sides are expanded as in \eqref{4.12}, with the appropriate
exponent and with ``Bernoulli" replaced by ``Euler" where appropriate.
Applying \eqref{4.10} then completes the proofs of both identities.
\end{proof}

\section{Some further identities}

In this final section we state and prove some further consequences of our
main results from Sections~2, 4 and 5, respectively.

\subsection{Consequences of Theorem~1}

In addition to the two identities in Corollary~4,
we can obtain one more consequence of Corollary~3 by multiplying both sides of
\eqref{2.9} by $a$ and then taking the limit as $a\rightarrow\infty$. Then all
the terms in the sum on the left disappear, with the exception of the $l=n-1$
term. On the right, the fraction in the sum tends to 1 for $l\geq 2$, and to 2
for $l=1$. Putting everything together, we get the following consequence.

\begin{corollary}
For integers $n\geq 1$ we have
\begin{equation}\label{2.13}
\sum_{l=0}^n\binom{n}{l}B_lB_{n-l}(x) = n(x-1)B_{n-1}(x)-(n-1)B_n(x).
\end{equation}
\end{corollary}

For $x=0$ this is Euler's identity \eqref{1.2}, but it is also a special case
of identity (5.11.2) in \cite{Ha}.

We can obtain even more consequences from Theorem~1 by setting
$a=b=\varepsilon$ and then taking the limit as $\varepsilon\rightarrow\infty$,
or by considering the terms in \eqref{2.5} as power series in $\varepsilon$.
We begin with the first case.

\begin{corollary}
For integers $n\geq 1$ we have
\begin{equation}\label{2.14}
\sum_{l=0}^n\binom{n}{l}\frac{1}{2^l}B_lB_{n-l}(x)
= \frac{n}{2^n}(2x-1)B_{n-1}(2x)-\frac{n-1}{2^n}B_n(2x)-\frac{n}{4}B_{n-1}(x).
\end{equation}
\end{corollary}

\begin{proof}
With $a=b=\varepsilon$, the following limits are obvious:
\[
\lim_{\varepsilon\rightarrow\infty}\frac{(\varepsilon)_l(\varepsilon)_{n-l}}{(2\varepsilon)_n}=\frac{1}{2^n},\qquad
\lim_{\varepsilon\rightarrow\infty}\frac{2\varepsilon(\varepsilon)_l}{(2\varepsilon)_{l+1}}=\frac{1}{2^l},\qquad
\lim_{\varepsilon\rightarrow\infty}\frac{\varepsilon^2}{(2\varepsilon+1)(2\varepsilon)}=\frac{1}{4}.
\]
Hence we have
\begin{equation}\label{2.15}
\frac{1}{2^n}\sum_{l=0}^n\binom{n}{l}B_l(x)B_{n-l}(x)
=\sum_{l=0}^n\binom{n}{l}\frac{1}{2^l}B_lB_{n-l}(x)+\frac{n}{4}B_{n-1}(x).
\end{equation}
The sum on the left has a well-known evaluation (see (50.11.2) in \cite{Ha}):
\[
\sum_{l=0}^n\binom{n}{l}B_l(x)B_{n-l}(x)=n(2x-1)B_{n-1}(2x)-(n-1)B_n(2x).
\]
This, with \eqref{2.15}, immediately gives \eqref{2.14}.
\end{proof}

We note that \eqref{2.15} can also be obtained as a special case of identity
(6.1) in \cite{AD}. For the next statement, again using
$a=b=\varepsilon$, we need the {\it second-order harmonic numbers\/}, defined
by $H_0^{(2)}:=0$ and
\[
H_n^{(2)}:=\sum_{j=1}^n\frac{1}{j^2}\qquad(n\geq 1).
\]
We can now prove the following result.

\begin{corollary}
For integers $n\geq 1$ we have
\begin{align}
\sum_{l=1}^{n-1}\left(H_{n-1}-H_{l-1}\right)
\frac{B_l(x)}{l}&\frac{B_{n-l}(x)}{n-l}
=\sum_{l=1}^{n}\binom{n}{l}\left(H_l+\frac{1}{l}\right)\frac{B_l}{l}B_{n-l}(x)\label{2.16} \\
&+nB_{n-1}(x)+\frac{1}{2}\left(H_{n-1}^2+3H_{n-1}^{(2)}\right)B_n(x).\nonumber
\end{align}
\end{corollary}

\begin{proof}
Setting $a=b=\varepsilon$ in \eqref{2.5} and dividing both sides by
$\varepsilon$, we have to expand the following terms. First, for
$1\leq l\leq n-1$ we have
\begin{align}
\frac{1}{\varepsilon}\frac{(a)_{l}(b)_{n-l}}{(a+b)_{n}}
&=\frac{(\varepsilon+1)\dots(\varepsilon+l-1)(\varepsilon+1)\dots(\varepsilon+n-l-1)}{2(2\varepsilon+1)\dots(2\varepsilon+n-1)}\label{2.17}\\
&=\frac{(l-1)!(n-l-1)!}{2(n-1)!}
\prod_{j=1}^{l-1}\left(1+\frac{\varepsilon}{j}\right)
\prod_{j=1}^{n-l-1}\left(1+\frac{\varepsilon}{j}\right)
\prod_{j=1}^{n-1}\left(1+\frac{2\varepsilon}{j}\right)^{-1} \nonumber\\
&=\frac{(l-1)!(n-l-1)!}{2(n-1)!}\left(1+\left(H_{l-1}+H_{n-l-1}-2H_{n-1}\right)\varepsilon+O(\varepsilon^2)\right).\nonumber
\end{align}
Next, for $l\geq 1$ we get
\begin{align}
\frac{1}{\varepsilon}\frac{a(b)_{l}+b(a)_{l}}{(a+b)_{l+1}}
&=\frac{(\varepsilon+1)\dots(\varepsilon+l-1)}{(2\varepsilon+1)\dots(2\varepsilon+l)}
=\frac{1}{l}\left(1+\frac{2\varepsilon}{l}\right)^{-1}
\prod_{j=1}^{l-1}\frac{1+\frac{\varepsilon}{j}}{1+\frac{2\varepsilon}{j}}\label{2.18}\\
&=\frac{1}{l}\left(1-\frac{2\varepsilon}{l}+\dots\right)
\prod_{j=1}^{l-1}\left(1-\frac{\varepsilon}{j}+\dots\right) \nonumber \\
&=\frac{1}{l}\left(1-\left(H_l+\frac{1}{l}\right)\varepsilon+O(\varepsilon^2)\right),\nonumber
\end{align}
where we have used the fact that $H_{l-1}+2/l=H_l+1/l$. Next, we have
\begin{equation}\label{2.19}
\frac{1}{\varepsilon}\frac{ab}{(a+b+1)(a+b)}=\frac{1}{2(1+2\varepsilon)}
=\frac{1}{2}\left(1-2\varepsilon+O(\varepsilon^2)\right).
\end{equation}
Finally, we collect on the right the terms left out in the two sums, namely
\[
\frac{1}{\varepsilon}\left(1-2\frac{(a)_n}{(a+b)_n}\right)B_n(x).
\]
We expand
\begin{align*}
2\frac{(a)_n}{(a+b)_n}&=\frac{2(\varepsilon)_n}{(2\varepsilon)_n}
=\prod_{j=1}^{n-1}\frac{\varepsilon+j}{2\varepsilon+j} \\
&=\prod_{j=1}^{n-1}\left(1+\frac{\varepsilon}{j}\right)\left(1+\frac{2\varepsilon}{j}\right)^{-1}
=\prod_{j=1}^{n-1}\left(1-\frac{\varepsilon}{j}+\frac{2\varepsilon^2}{j^2}+\dots\right) \\
&=1-H_{n-1}\varepsilon+\left(2H_{n-1}^{(2)}+\sum_{1\leq j<k\leq n-1}\frac{1}{jk}\right)\varepsilon^2+O(\varepsilon^3).
\end{align*}
Now the double sum in the last term can clearly be written as
$(H_{n-1}^2-H_{n-1}^{(2)})/2$, and thus
\begin{equation}\label{2.20}
\frac{1}{\varepsilon}\left(1-2\frac{(a)_n}{(a+b)_n}\right)
=H_{n-1}-\left(\frac{3}{2}H_{n-1}^{(2)}+\frac{1}{2}H_{n-1}^2\right)\varepsilon+O(\varepsilon^2).
\end{equation}
If we substitute \eqref{2.17}--\eqref{2.20} into \eqref{2.5} and let
$\varepsilon\rightarrow 0$, we recover the polynomial analogue of Miki's
identity. Finally, if we equate the coefficients of $\varepsilon$, we
immediately get \eqref{2.16} after multiplying both sides by $-1$ and
exploiting symmetry in \eqref{2.17}.
\end{proof}

\subsection{Consequences of Theorem~2}
We restrict our attention to the case $k=3$ and $a_1=a_2=a_3=\varepsilon$.
Furthermore, to avoid double indices, we set $i=l_1$, $j=l_2$, $l=l_3$ on the
left, and $i=l_0$, $j=l_1$, $l=l_2$ on the right of \eqref{4.0}. Then we get,
after dividing by $n!$,
\begin{gather}
\sum_{i+j+l=n}\frac{(\varepsilon)_i(\varepsilon)_j(\varepsilon)_l}
{(3\varepsilon)_n}\frac{B_i(x)}{i!}\frac{B_j(x)}{j!}\frac{B_l(x)}{l!}
=3\sum_{i+j+l=n}\frac{\varepsilon(\varepsilon)_j(\varepsilon)_l}
{(3\varepsilon)_{j+l+1}}\frac{B_i(x)}{i!}\frac{B_j}{j!}\frac{B_l}{l!}\label{6.9}\\
+3\sum_{i+j=n-1}\frac{\varepsilon^2(\varepsilon)_j}{(3\varepsilon)_{j+2}}
\frac{B_i(x)}{i!}\frac{B_j}{j!}
+\frac{\varepsilon^3}{(3\varepsilon)_3}\frac{B_{n-2}(x)}{(n-2)!},\nonumber
\end{gather}
valid for all $n\geq 2$ and $\varepsilon>0$. This will be the basis for the
various results in this subsection, and also immediately gives \eqref{4.0a}.

For a first easy consequence we let $\varepsilon\rightarrow\infty$ on both
sides of \eqref{6.9}. Then the limit of the four fractions involving
$\varepsilon$ are easily seen to be $3^{-n}$, $3^{-j-l-1}$, $3^{-j-2}$, and
$3^{-3}$, respectively. Thus, after multiplying both sides by $3^nn!$, we
immediately get the following result.

\begin{corollary}
For integers $n\geq 2$ we have
\begin{gather}
\sum_{i+j+l=n}\binom{n}{i,j,l}B_i(x)B_j(x)B_l(x)
=\sum_{i+j+l=n}\binom{n}{i,j,l}3^iB_i(x)B_jB_l\label{6.10} \\
+n\sum_{i=0}^{n-1}\binom{n-1}{i}3^iB_i(x)B_{n-1-i}+n(n-1)3^{n-3}B_{n-2}(x).\nonumber
\end{gather}
\end{corollary}

For the next consequence of \eqref{6.9} we set $x=0$, for greater simplicity
of the statement. The proof is tedious, and we leave the details to the
interested reader.

\begin{corollary}
For integers $n\geq 2$ we have
\begin{align*}
&\frac{1}{3}\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}
\frac{B_i}{i}\frac{B_j}{j}\frac{B_l}{l}
=\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i-1}
\frac{B_i}{i}\frac{B_j}{j}\frac{B_l}{l}
+\sum_{l=1}^{n-2}\binom{n-1}{l+1}\frac{B_l}{l}\frac{B_{n-l-1}}{n-l-1}\\
&+\sum_{l=1}^{n-1}\left(3H_{n-1}-2H_{l-1}+\tfrac{1}{n}\right)\frac{B_l}{l}\frac{B_{n-l}}{n-l}
-2\sum_{l=1}^{n-1}\binom{n-1}{l-1}\left(2H_l+\tfrac{1}{l}\right)\frac{B_l}{l}\frac{B_{n-l}}{n-l} \\
&+\frac{n-1}{6}B_{n-2}+\left(\frac{1}{(n-1)n}-3\right)B_{n-1}
-2\left(\frac{2}{n}H_{n-1}+H_{n-1}^2+2H_{n-1}^{(2)}+\frac{3}{n^2}\right)\frac{B_n}{n}.
\end{align*}
\end{corollary}
This can be seen as a third-order analogue of Miki's identity.
Note the difference in complexity between this result and the third-order
analogue of Matiyasevich's identity given in \eqref{4.0a}. See also
\cite[(6.5)]{AD} for a third-order ``Miki analogue" for Bernoulli polynomials.
To prove Corollary 9, one can use a similar method as in the proof of
Corollary~7, and proceed as follows:

-- Collect the ``edge" and ``corner" terms in the sums of \eqref{6.9}.

-- Divide both sides of \eqref{6.9} by $\varepsilon^2$.

-- Expand the various fractions involving $\varepsilon$ in a similar
way as in \eqref{2.17} and \eqref{2.18}.

-- Equate the constant terms (i.e., let $\varepsilon\rightarrow 0$) to
obtain a first identity.

-- Equate the coefficients of $\varepsilon$ to obtain a second identity.

\noindent
Interestingly, in this case the first identity turns out to be equivalent to
\eqref{1.3}, Miki's original identity. The second one is Corollary 9.

\subsection{Consequences of Theorems~3 and~4}

Given the similarities between Theorems~1 and~3, it is clear that Euler
analogues of Corollaries~2--7 could easily be derived; recall that an analogue
of Corollary~1 is already stated following Theorem~3. Here we restrict
ourselves to only a few more consequences; we also skip the proofs which are
again similar to the proof of Corollary~7.

\begin{corollary}
For integers $n\geq 2$ we have
\begin{align*}
\sum_{l=1}^{n-2}\frac{E_l(x)}{l}\frac{E_{n-l-1}(x)}{n-l-1}
&=4\sum_{l=1}^{n-1}\binom{n-2}{l-1}H_{l-1}\frac{B_{n-l}(x)}{n-l}\frac{E_l(0)}{l}\\
&\qquad\qquad+2H_{n-2}\frac{E_{n-1}(x)}{n-1}+4\frac{H_{n-1}}{n-1}\frac{E_n(0)}{n},
\end{align*}
and for $n\geq 1$,
\begin{align*}
&\sum_{l=1}^{n-1}\left(H_{n-1}-H_{l-1}\right)
\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
=\frac{1}{2}\left(H_{n-1}^2+3H_{n-1}^{(2)}\right)\frac{E_n(x)}{n} \\
&\quad+\sum_{l=1}^n\binom{n-1}{l-1}\left(H_{l-1}^2+3H_{l-1}^{(2)}\right)
\frac{B_{n+1-l}(x)}{n+1-l}\frac{E_l(0)}{l}
+\frac{1}{n}\left(H_n^2+3H_n^{(2)}\right)\frac{E_{n+1}(0)}{n+1}.
\end{align*}
\end{corollary}

Finally, to obtain third-order analogues of Miki's identity, we start with
\eqref{5.26} for $k=3$ and follow the outline described after Corollary~9.
This leads to the following identities.

\begin{corollary}
For integers $n\geq 2$ we have
\begin{align*}
\sum_{l=1}^{n-1}&\left(\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
-\frac{E_l(0)}{l}\frac{E_{n-l}(0)}{n-l}\right) \\
&\qquad\qquad=\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i}
E_i(x)\frac{E_j(0)}{j}\frac{E_l(0)}{l}+2H_{n-1}\frac{E_n(x)}{n},
\end{align*}
and
\begin{align*}
\frac{1}{3}&\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}
\frac{E_i(x)}{i}\frac{E_j(x)}{j}\frac{E_l(x)}{l}
=-2\left(H_{n-1}^2+2H_{n-1}^{(2)}\right)\frac{E_n(x)}{n} \\
&+\sum_{\substack{i+j+l=n\\i,j,l\geq 1}}\binom{n-1}{i}
\left(H_{j-1}+H_{l-1}-3H_{j+l-1}\right)E_i(x)\frac{E_j(0)}{j}\frac{E_l(0)}{l}\\
&+\sum_{l=1}^{n-1}\left(3H_{n-1}-H_{l-1}-H_{n-l-1}\right)
\left(\frac{E_l(x)}{l}\frac{E_{n-l}(x)}{n-l}
-\frac{E_l(0)}{l}\frac{E_{n-l}(0)}{n-l}\right).
\end{align*}
\end{corollary}

Using the special values \eqref{5.6}, numerous identities involving Genocchi
and/or Euler numbers could also be obtained.

\section{Final remarks}

{\bf 1.} By choosing different values of the parameters in Theorems~1--4,
many more identities for Bernoulli and Euler numbers and polynomials could be
obtained, some relatively simple, and others of increasing complexity.
We have shown in this paper that the original identities of Miki and
Matiyasevich and their various extensions are special cases of very general
class of identities.

\medskip
{\bf 2.} This is not the first common extension of the identities of Miki and
Matiyasevich. In fact, using generating functions, Dunne and Schubert \cite{DS}
recently proved the following result.

\begin{theorem}[Dunne and Schubert]
For any integer $n\geq 2$ and real $p\geq 0$ we have
\begin{align}
\sum_{l=1}^{n-1}(2l)_p(2n-2l)_p&\frac{B_{2l}}{2l}\frac{B_{2n-2l}}{2n-2l}
= 2B_{2n}\frac{\Gamma(2n+2p)}{(2n)!}\sum_{l=1}^{2n-1}
\frac{\Gamma(p+l)\Gamma(p+1)}{\Gamma(2p+l+1)}\label{7.1} \\
&+\frac{2\Gamma(p+1)}{(2n)!}\sum_{l=1}^n\binom{2n}{2l}
\frac{\Gamma(p+2l)\Gamma(2p+2n)}{\Gamma(2p+2l+1)}B_{2l}B_{2n-2l}.\nonumber
\end{align}
\end{theorem}

The case $p=0$ gives Miki's identity, while for $p=1$ we get
\[
\sum_{l=1}^nB_{2l}B_{2n-2l} = \frac{1}{n+1}
\sum_{l=1}^n\binom{2n+2}{2l+2}B_{2l}B_{2n-2l} + 2nB_{2n},
\]
which is equivalent to Matiyasevich's identity.

The identity \eqref{7.1} actually follows from Theorem~1 if we take $a=b=p$ and
$x=0$, then replace $n$ by $2n$ and extract the end terms in the sums. After
some manipulations we then get
\begin{align}
\sum_{l=1}^{n-1}(2l)_p(2n-2l)_p&\frac{B_{2l}}{2l}\frac{B_{2n-2l}}{2n-2l}
= \frac{\Gamma^2(p)}{(2n)!}\left((2p)_{2n}-2(p)_{2n}\right) \label{7.2} \\
&+\frac{2\Gamma(p+1)}{(2n)!}\sum_{l=1}^n\binom{2n}{2l}
\frac{\Gamma(p+2l)\Gamma(2p+2n)}{\Gamma(2p+2l+1)}B_{2l}B_{2n-2l}.\nonumber
\end{align}
Comparing \eqref{7.1} with \eqref{7.2} shows that after some simplification
we have
\[
\sum_{l=1}^{2n-1}\frac{\Gamma(p+l)}{\Gamma(2p+l+1)}
=\frac{\Gamma(p)}{\Gamma(2p+1)} - \frac{\Gamma(p+2n)}{p\Gamma(2p+2n)}.
\]
This last identity can be proved independently, for instance by manipulating
the integral representation of Euler's beta function.


text:
Q: Create multiple table using Microsoft interop word How can I  point to a specific location on a  active word document and insert multiple table after finding that specific location(location would be string).
public void Createtable()
    {
        object oMissing = System.Reflection.Missing.Value;
        object oEndOfDoc = "\\endofdoc"; /* \endofdoc is a predefined
                                                             bookmark */
        object oBookMark = "List of tool changer systems ";

        //Start Word and Open existing Safety device Word file document.
        Word._Application OpenWord;
        Word._Document OpenDoc;
        OpenWord = new Word.Application();
        OpenWord.Visible = true;
        OpenWord.WindowState = Word.WdWindowState.wdWindowStateMaximize;
        OpenDoc = OpenWord.Documents.Open(FilenametextBox.Text, ref oMissing, ref oMissing, ref oMissing);
        Word.Document wordDocument = OpenDoc as Word.Document;
        int tablecount = wordDocument.Tables.Count;
        wordDocument.Activate();

        int manualtablecount = 118;
        i = 0;
        for (int j = 1; j <= manualtablecount; j++)
        {
            Word.Table wTable = wordDocument.Tables[j];
            Word.Cell pCell = wTable.Cell(1, 1);
            Word.Cell pCell1 = wTable.Cell(2, 1);
            Word.Range wrdRng = wordDocument.Bookmarks.get_Item(ref oEndOfDoc).Range;

        if (pCell.Range.Text.Contains("List of tool changer systems"))
        {

          for (int iFW = 1; iFW <= 6; iFW++)
          {
            Word.Table FWTable;
            Word.Range wrdRangeFW = wordDocument.Bookmarks.get_Item(ref oBookMark).Range.Select();
            FWTable = wordDocument.Application.ActiveDocument.Tables.Add(wrdRangeFW,3, 5);

             FWTable.Range.ParagraphFormat.SpaceBefore = 3;
             FWTable.Range.Font.Size = 10;
             FWTable.Range.Font.Bold = 1;
             FWTable.Borders.Enable = 1;

              for (int r = 1; r <= 3; r++)
              {

                  if (r == 1)
                  {
                   FWTable.Range.ParagraphFormat.Alignment = Word.WdParagraphAlignment.wdAlignParagraphCenter;
                      FWTable.Cell(r, 1).Range.Text = "Designation";
                      FWTable.Cell(r, 2).Range.Text = "Sensor type";
                      FWTable.Cell(r, 3).Range.Text = "Modulename";
                      FWTable.Cell(r, 4).Range.Text = "Element";
                      FWTable.Cell(r, 5).Range.Text = "OK";
                    }
                    else
                    {
                      FWTable.Cell(r, c).Range.Text = "";
                    }

                    }
               }
          }
    }

Here i want to add six table after specific string available in the open word documents "List of tool changer systems"
I tried may thing but didnt get any success.
any help will be appreciated!!
Thanks

redpajama_set_name: RedPajamaStackExchange