  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. The present paper
can be viewed as an introduction to another our paper where the same approach
is applied to studying a dynamical model related to the z-measures (Markov
processes on partitions, Prob. Theory Rel. Fields 135 (2006), 84-152; arXiv:
math-ph/0409075).
