On a generalization of the Rogers generating function

• Published in: Journal of mathematical analysis and applications Vol. 475; no. 2; pp. 1019 - 1043
• Main Authors: Cohl, Howard S., Costas-Santos, Roberto S., Wakhare, Tanay V.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.07.2019
• Subjects: Basic hypergeometric orthogonal polynomials; Basic hypergeometric series; Connection coefficients; Definite integrals; Eigenfunction expansions; Generating functions; Eigenfunction expansions; Connection coefficients; Basic hypergeometric series; Basic hypergeometric orthogonal polynomials; Definite integrals; Generating functions; polynomials; Basic hypergeometric orthogonal
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2019.01.068

We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a ϕ12. 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 ϕ78, 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 ϕ78 to a ϕ12. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.