## Orthogonal polynomials and recurrence equations, operator equations and factorization

Wolfram Koepf

### Abstract

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or $q$-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class–implemented in the computer algebra system Maple–are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

Full Text (PDF) [206 KB]

### Key words

orthogonal polynomials, Hahn class, differential equations, difference equations, $q$-difference equations, hypergeometric functions, factorization of operator polynomials, computer algebra, Maple

### AMS subject classifications

33C45, 33C20, 33D45, 33D15, 39A70

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