2 00 8 Leonard pairs and the q - Racah polynomials ∗

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 diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V . In the appendix to [9] we outlined a correspondence between Leonard pairs and a class of orthogonal polynomials consisting of the q-Racah polynomials and some related polynomials of the Askey scheme. We also outlined how, for the polynomials in this class, the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be obtained in a uniform manner from the corresponding Leonard pair. The purpose of this paper is to provide proofs for the assertions which we made in that appendix. 1 Leonard pairs We begin by recalling the notion of a Leonard pair [5], [9], [10], [11], [12], [13], [14], [15], [16]. We will use the following terms. Let X denote a square matrix. Then X is called tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. Assume X is tridiagonal. Then X is called irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. We now define a Leonard pair. For the rest of this paper K will denote a field. Definition 1.1 [9] Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V , we mean an ordered pair of linear transformations A : V → V and A : V → V that satisfy both (i), (ii) below. (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. ∗