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Open AccessArticle

Some Generating Functions for q-Polynomials

1
Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA
2
Departamento de Física y Matemáticas, Facultad de Ciencias, Universidad de Alcalá, Alcalá de Henares, 28871 Madrid, Spain
3
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
*
Authors to whom correspondence should be addressed.
Symmetry 2018, 10(12), 758; https://doi.org/10.3390/sym10120758
Received: 14 November 2018 / Revised: 30 November 2018 / Accepted: 13 December 2018 / Published: 16 December 2018
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series 4 ϕ 5 , 5 ϕ 5 , 4 ϕ 3 , 3 ϕ 2 , 2 ϕ 1 , and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials. View Full-Text
Keywords: basic hypergeometric functions; generating functions; q-polynomials basic hypergeometric functions; generating functions; q-polynomials
MDPI and ACS Style

Cohl, H.S.; Costas-Santos, R.S.; Wakhare, T.V. Some Generating Functions for q-Polynomials. Symmetry 2018, 10, 758.

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