Charting the $q$-Askey 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.

arXiv comment17 pages, one figure; in v2 minor corrections, one remark and three references added; to appear in "Proceedings of the Conference on Hypergeometry, Integrability and Lie Theory", Contemporary Mathematics

References in corpus (9)

• A "missing" family of classical orthogonal polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Dunkl shift operators and Bannai-Ito polynomials
• Dualities in the $q$-Askey scheme and degenerate DAHA
• The Askey scheme as a four-manifold with corners
• Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials
• Limits of elliptic hypergeometric biorthogonal functions
• Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases
• Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Cited by in corpus (1)

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