Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2019; 34(4): 1175-1199

Online first article August 29, 2019      Printed October 31, 2019

https://doi.org/10.4134/CKMS.c180421

Copyright © The Korean Mathematical Society.

A $p$-deformed $q$-inverse pair and associated polynomials including Askey scheme

Rajesh V. Savalia

Charotar University of Science and Technology

Abstract

We construct a general bi-basic inverse series relation which provides extension to several $q$-polynomials including the Askey-Wilson polynomials and the $q$-Racah polynomials. We introduce a general class of polynomials suggested by this general inverse pair which would unify certain polynomials such as the $q$-extended Jacobi polynomials and $q$-Konhauser polynomials. We then emphasize on applications of the general inverse pair and obtain the generating function relations, summation formulas involving the associated polynomials and derive the $p$-deformation of some of the $q$-analogues of Riordan's classes of inverse series relations. We also illustrate the companion matrix corresponding to the general class of polynomials; this is followed by a chart showing the reducibility of the extended $p$-deformed Askey-Wilson polynomials as well as the extended $p$-deformed $q$-Racah polynomials.

Keywords: $q,p$-gamma function, $q,p$-Pochhammer symbol, $p$-deformed $q$-polynomial, $q$-inverse series

MSC numbers: Primary 33D15, 33D45, 33D65

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