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A \(p\)-deformed \(q\)-inverse pair and associated polynomials including Askey scheme. (English) Zbl 1429.33029

Summary: 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.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D65 Bibasic functions and multiple bases
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