The Askey Scheme for Hypergeometric Orthogonal Polynomials Viewed from Asymptotic Analysis

TL;DR

This paper explores asymptotic representations of hypergeometric orthogonal polynomials within the Askey scheme using generating functions, providing new insights into their limiting behaviors and relationships.

Contribution

It introduces a method to derive asymptotic representations of hypergeometric orthogonal polynomials via generating functions, connecting various polynomial families.

Findings

Asymptotic representations of Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials are derived.

Connections between these polynomials and Laguerre polynomials are established.

The approach enhances understanding of polynomial limits within the Askey scheme.

Abstract

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.