Fermionic representation for basic hypergeometric functions related to Schur polynomials

Abstract

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials considered by S.Milne \cite{Milne}. For $q=1$ these functions are also known as hypergeometric functions of matrix argument which are related to zonal spherical polynomials for $GL(N,C)/U(N)$ symmetric space. We show that these multivariable hypergeometric functions are tau-functions of the KP hierarchy. At the same time they are the ratios of Toda lattice tau-functions considered by Takasaki in \cite{Tinit}, \cite{T} evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type ${}_pF_s$ can be also viewed as group 2-cocycle for the $Ψ$DO on the circle of the order $p-s \leq 1$ (the group times are higher times of TL hierarchy and the arguments of hypergeometric function). We get the determinant representation and the integral representation of special type of KP tau-functions, these results generalize some of Milne's results in \cite{Milne}. We write down a system of linear differential and difference equations for these tau-functions (string equations). We present also fermionic representation for special type of Gelfand-Graev hypergeometric functions.