Difference system for Selberg correlation integrals

Abstract

The Selberg correlation integrals are averages of the products Πms=1 Πnl=1 (x s-z1)μs with respect to the Selberg density. Our interest is in the case m = 1, μ1 = μ, when this corresponds to the μth moment of the corresponding characteristic polynomial. We give the explicit form of an (n + 1) × (n + 1) matrix linear difference system in the variable ? which determines the average, and we give the Gauss decomposition of the corresponding (n + 1) × (n + 1) matrix. For μ a positive integer the difference system can be used to efficiently compute the power series defined by this average. © 2010 IOP Publishing Ltd.