On a generalization of the Rogers generating function

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},
}