An algebraic approach to the askey scheme of orthogonal polynomials

Abstract

Let K denote a field, and let V denote a vector space over K with finite. positive dimension. We consider a pair of linear transformations A : V → V and A?: V → V that satisfy the following two conditions: 1. There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A? is diagonal. 2. There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V . We give a correspondence between Leonard pairs and a class of orthogonal polynomials. This class coincides with the terminating branch of the Askey scheme and consists of the q-Racah, q-Hahn, dual q-Hahn q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials. We describe the above correspondence in detail. We show how, for the listed polynomials the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be expressed in a uniform and attractive manner using the corresponding Leonard pair. We give some examples that indicate how Leonard pairs arise in representation theory and algebraic combinatorics. We discuss a mild generalization of a Leonard pair called a tridiagonal pair. At the end we list some open problems. Throughout these notes our argument is elementary and uses only linear algebra. No prior exposure to the topic is assumed.