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

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.