  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.
