The power collection method for connection relations: Meixner polynomials.

*(English)*Zbl 1424.33020Summary: 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.

##### MSC:

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, \({}_pF_q\) |

##### Keywords:

generating functions; connection coefficients; connection-type relations; eigenfunction expansions; definite integrals; infinite series##### Software:

DLMF##### References:

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