A $p$-deformed $q$-inverse pair and associated polynomials including Askey scheme
Commun. Korean Math. Soc. 2019 Vol. 34, No. 4, 1175-1199
Published online October 31, 2019
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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