Charting the q-Askey Scheme. III. Verde-Star Scheme for  \(q=1\)

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.