  We consider a point process on one-dimensional lattice originated from the
harmonic analysis on the infinite symmetric group, and defined by the
z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian
formula for the correlation function of this process. Namely, we prove that the
correlation function is given as a Pfaffian with a matrix kernel. The kernel is
given in terms of the Gauss hypergeometric functions, and can be considered as
a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G.
Olshanski. Our result holds for all values of admissible complex parameters.
