2 7 A ug 2 00 4 Two linear transformations each tridiagonal with respect to an eigenbasis of the other ; an algebraic approach to the Askey scheme of orthogonal polynomials ∗ Paul Terwilliger

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: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) 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, qKrawtchouk, 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. ∗Lecture notes for the summer school on orthogonal polynomials and special functions, Universidad Carlos III de Madrid, Leganes, Spain. July 8–July 18, 2004. http://www.uc3m.es/uc3m/dpto/MATEM/summerschool/indice.html