Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

TL;DR

This paper explores the connection between hypergeometric discrete multiple orthogonal polynomials, tau functions, and integrable systems, revealing new solutions to Toda-type equations and characterizing their recurrence relations.

Contribution

It introduces tau function representations for these polynomials, links them to integrable equations, and characterizes Laguerre-Freud equations for specific polynomial families.

Findings

Tau functions expressed as double Wronskians of hypergeometric series.

Solutions to multicomponent Toda and Nijhoff-Capel equations.

Recursion coefficients satisfy new nonlinear Laguerre-Freud equations.

Abstract

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of tau functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff-Capel totally discrete Toda equations. The hypergeometric $\tau$-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre-Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.