Charting the $q$-Askey scheme

TL;DR

This paper provides a detailed classification and parametrization of orthogonal polynomial families within the $q$-Askey scheme, using Laurent polynomial sequences and a graphical scheme to illustrate their relationships and limit transitions.

Contribution

It introduces a new parametrization method for $q$-orthogonal polynomials using Laurent polynomial sequences and describes the underlying four-manifold structure of the scheme.

Findings

Classification and parametrization of $q$-orthogonal polynomial families.

Graphical representation of the $q$-Askey scheme and limit transitions.

Identification of the four-manifold structure underlying the scheme.

Abstract

Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q$-Askey scheme together with their $q$-hypergeometric representations by three sequences $x_k, h_k, g_k$ of Laurent polynomials in $q^k$, two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme.