  We study a 3-parametric family of stochastic point processes on the
one-dimensional lattice originated from a remarkable family of representations
of the infinite symmetric group. We prove that the correlation functions of the
processes are given by determinantal formulas with a certain kernel. The kernel
can be expressed through the Gauss hypergeometric function; we call it the
hypergeometric kernel.
  In a scaling limit our processes approximate the processes describing the
decomposition of representations mentioned above into irreducibles. As we
showed before, see math.RT/9810015, the correlation functions of these limit
processes also have determinantal form with so-called Whittaker kernel. We show
that the scaling limit of the hypergeometric kernel is the Whittaker kernel.
  The integral operator corresponding to the Whittaker kernel is an integrable
operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the
hypergeometric kernel can be considered as a kernel defining a `discrete
integrable operator'.
  We also show that the hypergeometric kernel degenerates for certain values of
parameters to the Christoffel-Darboux kernel for Meixner orthogonal
polynomials. This fact is parallel to the degeneration of the Whittaker kernel
to the Christoffel-Darboux kernel for Laguerre polynomials.
