Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

TL;DR

This paper extends Burchnall's method to derive explicit expansion formulas for a wide class of orthogonal polynomials within the Askey scheme, including their modifications and applications to the Toda lattice.

Contribution

It generalizes Burchnall's identities to all polynomial families in the Askey scheme and provides explicit expansions for modified weights and Toda lattice solutions.

Findings

Explicit expansion formulas for Wilson and Askey-Wilson polynomials.

New expansions for classical polynomials with modified weights.

Connections between orthogonal polynomial expansions and Toda lattice solutions.

Abstract

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.