Charting the q-Askey scheme. III. Verde-Star scheme for q=1 · arXivDesk
2307.06668Jul 13, 2023v2: minor corrections, 16 pages, 2 figures; to appear in "Orthogonal polynomials and special functions", K. Castillo and A.J. Durán (eds.), Springer-Verlag (volume dedicated to the memory of José Carlos Petronilho)
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 xk,hk,gk
Nearby in the stack
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