Difference system for Selberg correlation integrals

The Selberg correlation integrals are averages of the products ${\prod }_{s=1}^m{\prod }_{l=1}^n(x_s-z_l{)}^{{\mu }_s}$ 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.