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

**Delft University of Technology**

Faculty of Information Technology and Systems

Department of Technical Mathematics and Informatics

Report no. 98-17

1998

## Abstract

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials
and we give a *q*-analogue of this scheme containing basic hypergeometric
orthogonal polynomials.

In chapter 1 we give the definition, the orthogonality relation, the three term
recurrence relation, the second order differential or difference equation, the
forward and backward shift operator, the Rodrigues-type formula and generating
functions of all classes of orthogonal polynomials in this scheme.

In chapter 2 we give the limit relations 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, second order difference equation, forward and backward shift
operator, Rodrigues-type formula 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.

## Acknowledgement

We would like to thank Professor Tom H. Koornwinder who suggested us to
write a report like this. He also helped us solving many problems we
encountered during the research and provided us with several references.

Last modified on **March 30, 2001**