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# Paper: On a generalization of the Rogers generating function

## On a generalization of the Rogers generating function

Cohl, H. S., Costas-Santos, R. S., and Ge, L. Symmetry 2, no. 8 (2020), 1290

ACADEMIC PRESS INC ELSEVIER SCIENCE (USA) | ISSN: 0022-247X | JCR® 2019 Impact Factor: 1.220 - MATHEMATICS — position: 77/325 (Q1/T1)

## 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.

489 KB Preprint (PDF, 27 Pages)

## 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},
}
}