Difference system for Selberg correlation integrals

The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{μ_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $μ_1 = μ$, when this corresponds to the $μ$-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 $μ$ which determines the average, and we give the Gauss decomposition of the corresponding $(n+1) \times (n+1)$ matrix. For $μ$ a positive integer the difference system can be used to efficiently compute the power series defined by this average.

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