The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

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 2×2 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 (2000) [5]. Our result holds for all values of admissible complex parameters.