Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble

TL;DR

This paper generalizes the calculation of joint moments of characteristic polynomials and their derivatives for the circular β-ensemble, connecting to hypergeometric functions and Jack polynomials, with explicit formulas for all β > 0.

Contribution

It extends previous results to a β-general setting using Jack polynomials, providing explicit evaluations of scaled moments for all β > 0 under certain conditions.

Findings

Explicit formulas for joint moments for all β > 0.

Connection between moments and hypergeometric functions based on Jack polynomials.

Analysis of moments of the singular statistic in the Jacobi 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 relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a $\beta$-generalisation of these results, where now the role of the Schur polynomials is played by the Jack polynomials. This leads to an explicit evaluation of the scaled moments for all $\beta > 0$, subject to the constraint that a particular parameter therein is equal to a non negative integer. Consideration is also given to the calculation of the moments of the singular statistic $\sum_{j=1}^N 1/x_j$ for the Jacobi $\beta$-ensemble.