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