n-Kernel orthogonal polynomials on the Dirichlet, Dirichlet-Multinomial, Poisson-Dirichlet and Ewens sampling distributions, and positive-definite sequences

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 first half of the paper is devoted to providing a full characterization and probabilistic interpretation of |n|-orthogonal polynomial kernels (i.e. sums of products of orthogonal polynomials of the same degree |n|) 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 estab- lish 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 -> ∞