Difference system for Selberg correlation integrals

TL;DR

This paper derives an explicit matrix difference system for Selberg correlation integrals, particularly for the case related to characteristic polynomial moments, enabling efficient computation for integer parameters.

Contribution

It introduces a new explicit matrix difference system for Selberg correlation integrals in the case m=1, providing a computational tool for moments of characteristic polynomials.

Findings

Explicit $(n+1) imes (n+1)$ matrix difference system derived.

Gauss decomposition of the matrix provided.

Efficient computation of the average for positive integer $$.

Abstract

The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{\mu_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $\mu_1 = \mu$, when this corresponds to the $\mu$-th moment of the corresponding characteristic polynomial. We give the explicit form of a $(n+1) \times (n+1)$ matrix linear difference system in the variable $\mu$ which determines the average, and we give the Gauss decomposition of the corresponding $(n+1) \times (n+1)$ matrix. For $\mu$ a positive integer the difference system can be used to efficiently compute the power series defined by this average.