Quadratic transformations for orthogonal polynomials in one and two variables

TL;DR

This paper explores quadratic transformations among orthogonal polynomials in one and two variables, detailing their relationships within the Askey and $q$-Askey schemes, and focusing on $BC_2$-type polynomials.

Contribution

It provides a comprehensive list of quadratic transformations in the one-variable case and analyzes specific two-variable polynomials related to root system $BC_2$.

Findings

Enumerates quadratic transformations in the Askey scheme

Identifies transformations for $BC_2$-type Jacobi and Koornwinder polynomials

Highlights structural relationships among multivariable orthogonal polynomials

Abstract

We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system $BC_2$, i.e., $BC_2$-type Jacobi polynomials if $q=1$ and Koornwinder polynomials in two variables in the $q$-case.