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New forms of the Cauchy operator and some of their applications. (English) Zbl 1342.33038

Summary: In this paper, we first construct the Cauchy \(q\)-shift operator \(T(a,b;D_{xy})\) and the Cauchy \(q\)-difference operator \(L(a,b;\theta_{xy})\). We then apply these operators in order to represent and investigate some new families of \(q\)-polynomials which are defined in this paper. We derive some \(q\)-identities such as generating functions, symmetry properties and Rogers-type formulas for these \(q\)-polynomials. We also give an application for the \(q\)-exponential operator \(R(bD_q)\).

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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