## On a generalization of the Rogers generating function

Cohl, H. S., Costas-Santos, R. S., and Wakhare, T. V.**Journal of Mathematical Analysis and Applications**475, no. 2 (2019), 1019 — 1043

## Abstract

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 \({}_8\phi_7\), 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.

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## BibTeX

@article {MR3944361, AUTHOR = {Cohl, H. S. and Costas-Santos, Roberto S. and Wakhare, Tanay V.}, TITLE = {On a generalization of the {R}ogers generating function}, JOURNAL = {J. Math. Anal. Appl.}, FJOURNAL = {Journal of Mathematical Analysis and Applications}, VOLUME = {475}, YEAR = {2019}, NUMBER = {2}, PAGES = {1019--1043}, ISSN = {0022-247X}, MRCLASS = {33D05}, MRNUMBER = {3944361}, DOI = {10.1016/j.jmaa.2019.01.068}, URL = {https://doi.org/10.1016/j.jmaa.2019.01.068}, }