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The power collection method for connection relations: Meixner polynomials. (English) Zbl 1424.33020
Summary: We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the \((q)\)-Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials. These relations are then used to derive generalizations of generating functions for these orthogonal polynomials. The coefficients of these generalized generating functions are in general, given in terms of multiple hypergeometric functions. From derived generalized generating functions, we deduce corresponding contour integral and infinite series expressions by using orthogonality.
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05A15 Exact enumeration problems, generating functions
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
33C20 Generalized hypergeometric series, \({}_pF_q\)
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