Leonard pairs and the q-Racah 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: (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. Wecall such a pair a Leonard pair on V. In the appendix to [Linear Algebra Appl. 330 (2001), p. 149] 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.