The intertwining property for $\beta $-Laguerre processes and integral operators for Jack polynomials

TL;DR

This paper establishes new intertwining relations for $eta$-Laguerre processes using a novel Markov kernel linked to Jack polynomials, leading to integral formulas for multivariate Laguerre and hypergeometric functions.

Contribution

Introduces a Markov kernel depending on $eta$ and $\alpha$ that reveals intertwining relations for $eta$-Laguerre processes and derives integral formulas for related multivariate functions.

Findings

Jack polynomials are eigenfunctions of the Markov kernel

New intertwining relations for $eta$-Laguerre processes

Derived integral formulas for multivariate Laguerre and hypergeometric functions

Abstract

The aim of this paper is to study intertwining relations for Laguerre process with inverse temperature $\beta \ge 1$ and parameter $\alpha >-1$. We introduce a Markov kernel that depends on both $\beta $ and $ \alpha $, and establish new intertwining relations for the $\beta$-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.