Duality of Orthogonal Polynomials on a Finite Set

We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial ensemble. Introduction This note is about a certain duality of orthogonal polynomials defined on a finite set. If the weights of two systems of orthogonal polynomials are related in a certain way, then the values of the nth polynomial of the first system at the points of the set equal, up to a simple factor, the corresponding values of the (M −n)th polynomial of the second system, where M is the cardinality of the underlying finite set. We formulate the exact result and prove it in §1. In §2 we explain the motivation which led to the result. We compare two different ways to compute probabilistic quantities called correlation functions in a certain model. The model is a discrete analog of the orthogonal polynomial ensembles which appeared for the first time in the random matrix theory, see, e.g., [Dy], [Ga], [GM], [Me], [NW]. Discrete orthogonal polynomial ensembles were discussed in [BO1], [BO2] [BO3], [J1]-[J3]. The results of the two computations must be equal, but this is not at all obvious from the explicit formulas. Our duality relation provides a proof of the equivalence of the two resulting expressions. In §3 we consider 2 examples when the orthogonal polynomials are classical (Krawtchouk and Hahn polynomials). In these cases the duality provides relations between similar polynomials with different sets of parameters. The relations are also easily verified using known explicit formulas for the polynomials. I am very grateful to Grigori Olshanski for numerous discussions. I also want to thank Tom Koornwinder for providing me with his computation regarding the Hahn polynomials, see §3. 1. Duality Theorem 1. Let X = {x0, x1, . . . , xM} ⊂ R be a finite set of distinct points on the real line, u(x) and v(x) be two positive functions on X such that u(xk)v(xk) = 1 ∏ i6=k(xk − xi) 2 , k = 0, 1, . . . ,M, (1) Typeset by AMS-TEX 1