A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials

TL;DR

This paper derives asymptotic formulas for averages of derivatives of characteristic polynomials in random matrix groups, linking them to moments of derivatives of L-functions in number theory, and introduces a recurrence relation for these constants.

Contribution

It introduces a differential recurrence relation for determinants of hypergeometric functions related to moments of derivatives of characteristic polynomials, enabling efficient computation.

Findings

Derived asymptotic formulas for matrix averages over USp(2N), SO(2N), and O^-(2N).

Established a Toda lattice-like recurrence for constants in the formulas.

Connected random matrix results to moments of derivatives of L-functions.

Abstract

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of \tau-functions associated with Painlev\'e differential equations.