Orthogonality Relations for Multivariate Krawtchouk Polynomials

The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Gr\"unbaum and Rahman in [SIGMA 6 (2010), 090, 12 pages, arXiv:1007.4327]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.


Introduction
Consider where N 0 is the set of nonnegative integers and |x| = x 0 + x 1 + · · · + x n−1 . We recall the multinomial coefficient N x = N x 0 , . . . , x n−1 = (−1) x 1 +···+x n−1 (−N ) x 1 +···+x n−1 x 1 ! · · · x n−1 ! for x ∈ X(n, N ). Let M n (R) be the set of all n × n matrices over a set R. We fix x ∈ X(n, N ) and A = (a ij ) 1≤i,j≤n−1 ∈ M n−1 (C). We define the functions φ A (x; m) of m = (m 0 , . . . , m n−1 ) ∈ X(n, N ) by the following generating function where t m = t m 0 0 t m 1 1 · · · t m n−1 n−1 and a 0j = a i0 = 1 for 0 ≤ i, j ≤ n − 1. We know a hypergeometric expression of φ A (x; m) where M n−1 (N 0 ) is the set of square matrices of degree n − 1 with nonnegative integer elements. We prove the formula (2) in the last section of this paper.
This type of hypergeometric functions was originally defined by Aomoto and Gel'fand for general parameters. We are interested in the aspects of discrete orthogonal polynomials of these functions with weights . For a special case, when n = 2, they are well known and called the Krawtchouk polynomials. For general values of n, the author shows that such orthogonal polynomials appear as the zonal spherical functions of Gel'fand pairs of complex reflection groups [3]. In general, the author and H. Tanaka give the orthogonality relation of φ A (x; m)s by using the character algebras [4]. R.C. Griffiths shows that the polynomials defined by the generating function (1) are mutually orthogonal [1]. In their paper [2], Grünbaum and Rahman discuss and determine the necessary and sufficient conditions of the orthogonality of φ A (x; m)s for A ∈ M 2 (C), which are proved by analytic methods. In these four literature, the authors consider the case that weights b n (x; N ; η (i) ) are positive.
In this study, we give a linear algebraic proof of the Grünbaum and Rahman's condition for general values of n and for arbitrary weights including the complex case. Our proof of the sufficient condition is close to [1] and [4]. For . Our main result is as follows: Theorem 1. The following are equivalent.

(b) A relation
holds for some N th root of unity ζ. Here A * 0 is the conjugate transpose of A 0 and D i = diag(η 0i , η 1i , . . . , η n−1i ) ∈ GL n (C) is a diagonal matrix (i = 1, 2). Remark 1. We assume that the diagonal elements of D 1 and D 2 appearing in the abovementioned theorem are real. Thus, one can recover the formula (1.18) of Grünbaum and Rahman's paper [2] by substituting n = 3 and ). We assume that a pair of finite groups (G, H) is a Gel'fand pair and A 0 is the table of the zonal spherical functions of (G, H). Let D 0 , . . . , D n−1 be the double cosets of H in G, and d 0 , . . . , d n be the dimensions of the irreducible components of 1 G H . Put D 1 = diag(|D 0 |, . . . , |D n−1 |) and D 2 = diag(|G|/d 0 , . . . , |G|/d n−1 ). Then (3) holds from the orthogonality relation of the zonal spherical functions. Furthermore φ A (x; m)'s are realized as the zonal spherical functions of a Gel'fand pair (G ≀ S N , H ≀ S N ). Therefore they satisfy the orthogonality relation (a) in the theorem. In general, A 0 is an eigenmatrix of a character algebra is considered in [4]. This paper organized as follows. First, we prove the main theorem in the next section. Second, we prove (2) in the last section. It seems to be the first explicit proof of this fact.

Proof of (2)
Here we give a proof of the formula (2) through direct computations. We need the following lemma.