The power collection method for connection relations: Meixner polynomials

TL;DR

This paper introduces the power collection method for deriving connection relations in hypergeometric orthogonal polynomials, exemplified on Meixner and Krawtchouk polynomials, leading to generalized generating functions and integral representations.

Contribution

The paper presents a new systematic method, the power collection method, for deriving connection relations for hypergeometric orthogonal polynomials within the (q-)Askey scheme.

Findings

Derived connection and connection-type relations for Meixner and Krawtchouk polynomials.

Obtained generalized generating functions with coefficients expressed as multiple hypergeometric functions.

Established contour integral and infinite series representations from the generalized generating functions.

Abstract

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 term of multiple hypergeometric functions. From derived generalized generating functions, we derive corresponding contour integral and infinite series expressions by using orthogonality.