Connection coefficients for classical orthogonal polynomials of several variables

TL;DR

This paper explores connection coefficients between different bases of classical multivariable orthogonal polynomials, revealing their relationships with Racah polynomials and providing explicit formulas for various polynomial families.

Contribution

It introduces new explicit formulas for connection coefficients of multivariable orthogonal polynomials, linking them to Racah polynomials and uncovering dualities and bispectral properties.

Findings

Connection coefficients for Jacobi polynomials relate to multivariable Racah polynomials.

Explicit formulas for Hahn and Krawtchouk polynomial connection coefficients are derived.

New interpretations of duality and bispectrality in multivariable orthogonal polynomials are provided.

Abstract

Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.