On a generalization of the Rogers generating function

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:475

Article

Cohl, Howard S.¹  Costas-Santos, Roberto S.²  Wakhare, Tanay V.³ 

[1] NIST, Appl & Computat Math Div, Mission Viejo, CA 92694 USA

[2] Univ Alcala De Henares, Dept Fis & Matemat, Alcala De Henares 28871, Spain

[3] Univ Maryland, Dept Math, College Pk, MD 20742 USA

关键词: Basic hypergeometric series;    Basic hypergeometric orthogonal polynomials;    Generating functions;    Connection coefficients;    Eigenfunction expansions;    Definite integrals;   

DOI  :  10.1016/j.jmaa.2019.01.068

来源: Elsevier

【 摘 要 】

We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a 2 phi 1. From that expansion, we derive corresponding specialization and limit transition expansions for the continuous q-Hermite, continuous q-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey Wilson polynomials by Ismail & Simeonov whose coefficient is a 807, we derive corresponding generalized expansions for the Wilson, continuous q-Jacobi, and Jacobi polynomials. By comparing the coefficients of the Askey Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric functions which relates an 8 phi 7 to a 2 phi 1. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality. Published by Elsevier Inc.

【 授权许可】

Free