JOINT MOMENTS OF A CHARACTERISTIC POLYNOMIAL AND ITS DERIVATIVE FOR THE CIRCULAR β-ENSEMBLE

Abstract

The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble and furthermore that they relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a β-generalisation of these results, where now the role of the Schur polynomials is played by the Jack poly..