Multivariate hypergeometric functions as τ-functions of Toda lattice and Kadomtsev–Petviashvili equation

We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as τ-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. The role of additional symmetries in generating hypergeometric τ-functions is explained.