# Paper No. 10-07

**RC Griffiths and D Spano**

**n-Kernel Orthogonal Polynomials on the Dirichlet, Dirichlet-Multinomial, Poisson-Dirichlet and Ewens sampling distributions, and positive-definite sequences**

**Abstract:** We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coe±cients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet-Multinomial distribution, respectively on the continuous and the *N*-discrete *d*-dimensional simplex. Their infinite-dimensional limit distributions, respectively the Poisson-Dirichlet distribution and the Ewens' sampling formula, are considered as well. We study in particular the possibility of mapping canonical correlations on the *d*-dimensional continuous simplex (i) to canonical correlation sequences on the *d *+ 1-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and viceversa. Driven by this motivation, the ¯rst half of the paper is devoted to providing a full characterization and probabilistic interpretation of *j**n**j*-orthogonal polynomial kernels (i.e. sums of products of orthogonal polynomials of the same degree *j**n**j*) with respect to the mentioned marginal distributions. Orthogonal polynomial kernels are important to overcome some non-uniqueness di±culties arising when dealing with multivariate orthogonal (or bi-orthogonal) polynomials.We establish several identities and some integral representations which are multivariate extensions of important results known for the case *d *= 2 since the 1970's. These results, along with a common interpretation of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to several non-trivial solutions to Lancaster's problem, many of which can be extended naturally to the limit as *d **! 1**:* AMS 2000 subject classifications: 33C50, 60E05, 60G07.

**Keywords: **Multivariate orthogonal polynomials, Orthogonal polynomial kernels, Jacobi, Hahn, Dirichlet distribution, Dirichlet-Multinomial, Poisson-Dirichlet, Ewens' sampling formula, Canonical correlations, Positive-de¯nite sequences, Lancaster's problem.