2025/08/11 by Howard S. Cohl, Cohl, Howard S., Roberto S. Costas-Santos +3
Mathematics · #33D15 #33D45 #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2508.08162
openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the q and q-1-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly setting one of the free parameters (not q) equal to zero until no parameters are left. These subfamilies (and their q-1 counterparts) are the continuous dual q-Hahn, Al-Salam--Chihara, continuous big q-Hermite, and the continuous q-Hermite polynomials. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulas which these polynomials satisfy. We then study the terminating transformation structure which are implied by the terminating representations of these polynomials. We conclude by describing the symmetry group structure of the q-Askey scheme.