  The present work stemmed from the study of the problem of harmonic analysis
on the infinite-dimensional unitary group U(\infty). That problem consisted in
the decomposition of a certain 4-parameter family of unitary representations,
which replace the nonexisting two-sided regular representation (Olshanski, J.
Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by
certain probability measures on an infinite-dimensional space \Omega, which is
a dual object to U(\infty). A way to describe those measures is to convert them
into determinantal point processes on the real line, it turned out that their
correlation kernels are computable in explicit form --- they admit a closed
expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and
Olshanski, Ann. Math., 2005, arXiv:0109194).
  In the present work we describe a (nonevident) q-discretization of the whole
construction. This leads us to a new family of determinantal point processes.
We reveal its connection with an exotic finite system of q-discrete orthogonal
polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point
processes live on a double q-lattice and we show that their correlation kernels
are expressed through the basic hypergeometric function 2-\phi-1.
  A crucial novel ingredient of our approach is an extended version G of the
Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin
branching rule for irreducible representations of unitary groups). We find the
q-boundary of G, thus extending previously known results (Gorin, Adv. Math.,
2012, arXiv:1011.1769).
