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

Combinatorics