The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

TL;DR

This paper comprehensively catalogs the Askey-scheme of hypergeometric orthogonal polynomials, detailing their properties, limit relations, and q-analogues, providing a structured overview of classical and basic hypergeometric polynomials.

Contribution

It systematically presents the definitions, relations, and q-analogues of all classes within the Askey-scheme, including their limit relations and derivations from q-analogues.

Findings

Complete listing of orthogonality relations and recurrence relations.

Explicit limit relations between different polynomial classes.

Connection between classical and q-analogues of hypergeometric polynomials.

Abstract

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.