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

TL;DR

This paper establishes a connection between q-deformed multivariate hypergeometric functions and tau-functions of the KP and Toda lattice hierarchies, revealing their role in integrable systems and symmetries.

Contribution

It introduces a novel link between hypergeometric functions and integrable hierarchies, showing how these functions serve as tau-functions with variables as higher times.

Findings

Hypergeometric functions are tau-functions of KP and Toda hierarchies.

Higher times of hierarchies correspond to hypergeometric function variables.

Additional symmetries generate hypergeometric tau-functions.

Abstract

We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as tau-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 tau-functions is explained.