The intertwining property for ${\hat{I}}^2$-Laguerre processes and integral operators for Jack polynomials

Abstract

The aim of this paper is to study intertwining relations for Laguerre process with inverse temperature ${\hat{I}}^2\ge 1$ and parameter $\hat{I}\pm >-1$. We introduce a Markov kernel that depends on both ${\hat{I}}^2$ and $\hat{I}\pm$, and establish new intertwining relations for the ${\hat{I}}^2$-Laguerre processes using this kernel. A key observation is that Jack symmetric polynomials are eigenfunctions of our Markov kernel, which allows us to apply a method established by Ramanan and Shkolnikov. Additionally, as a by-product, we derive an integral formula for multivariate Laguerre polynomials and multivariate hypergeometric functions associated with Jack polynomials.

15 pages