Explicit description of Christoffel deformations and Palm measures of the Plancherel measure, the z-measures and the Gamma process

Abstract

The Christoffel deformation of a measure on the real line consists of multiplying this measure by a squared polynomial having its roots in $R$. We introduce Christoffel deformations of discrete orthogonal polynomial ensembles by considering the Christoffel deformations of the underlying measure, and prove that this construction extends to more general point processes describing distributions on partitions: the poissonized Plancherel measure and the z-measures. These deformations contain the theory of Palm measures, and as a consequence, we obtain explicit formulas for the Palm measures of the poissonized Plancherel measure and the z-measures in terms of discrete Wronskians in the relevant special functions. The extension to the Plancherel measure is obtained via a limit transition from the Charlier ensemble, while the extension to the z-measures follows from an analytic continuation argument. A limit procedure starting from the non-degenerate z-measures leads to a deformation of the Gamma process introduced by Borodin and Olshanski.