Pearson Equations for Discrete Orthogonal Polynomials: I. Generalized Hypergeometric Functions and Toda Equations

TL;DR

This paper explores the algebraic and differential equations governing semiclassical discrete orthogonal polynomials, revealing connections to hypergeometric functions, Toda lattices, and integrable systems, with new insights into their structure and symmetries.

Contribution

It introduces a banded Laguerre-Freud matrix for semiclassical polynomials and links hypergeometric function relations to symmetries in the moment matrix, advancing understanding of their integrable structures.

Findings

Laguerre-Freud matrix is banded in the semiclassical case

Contiguous relations for hypergeometric functions translate into symmetries of the moment matrix

Discrete Toda lattice describes shifts in squared norms of orthogonal polynomials

Abstract

The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $\pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.