Meixner polynomials and random partitions

The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the discrete hypergeometric kernel by a new method, based on a relationship between the z-measures and the Meixner orthogonal polynomial ensemble. In another paper (Prob. Theory Rel. Fields 135 (2006), 84–152) we apply the same approach to a dynamical model related to the z-measures.