Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$

TL;DR

This paper provides a detailed classification and uniform parametrization of orthogonal polynomial families in the $q=1$ Askey scheme, extending Verde-Star's framework and illustrating limit transitions from the $q$-case.

Contribution

It introduces a systematic labeling and parametrization of $q=1$ orthogonal polynomials, enhancing understanding of their structure and connections within the Askey scheme.

Findings

Unified parametrization of orthogonal polynomial families

Graphical representation of limit transitions

Explicit classification excluding Hermite polynomials

Abstract

Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way.