Invariant properties for Wronskian type determinants of classical and classical discrete orthogonal polynomials under an involution of sets of positive integers

TL;DR

This paper uncovers an invariant property of Casorati and Wronskian determinants involving classical discrete and continuous orthogonal polynomials, revealing symmetries related to difference equations and Christoffel transforms.

Contribution

It introduces a novel symmetry property of determinants of orthogonal polynomials under a specific set involution, extending to continuous cases via limits.

Findings

Invariant property of Casorati determinants under set involution

Extension of invariance to Wronskian determinants of Hermite, Laguerre, Jacobi polynomials

Connection between invariance and higher order difference equations

Abstract

Given a finite set $F=\{f_1,\cdots ,f_k\}$ of nonnegative integers (written in increasing size) and a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant $\det(p_{f_i}(x+j-1))_{i,j=1,\cdots,k}$. In this paper we prove a nice invariant property for this kind of Casorati determinants when the set $F$ is changed by $I(F)=\{0,1,2,\cdots,\max F\}\setminus \{\max F-f:f\in F\}$. This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Christoffel transforms of the classical discrete measures. By passing to the limit, this invariant property is extended for Wronskian type determinants whose entries are Hermite, Laguerre and Jacobi polynomials.