Scalar Products of Symmetric Functions and Matrix Integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions, and integrals defining matrix-model partition functions. Using the fermionic Fock space representation, we prove an expansion of an associated class of KP and 2-Toda tau functions $\tau_{r,n}$ in a series of Schur functions generalizing the hypergeometric series and relate it to the scalar product formulas. We show how special cases of such tau functions can be identified as formal series for partition functions. A closed form expansion of $\ln\tau_{r,n}$ in terms of Schur functions is derived.