A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials
Number Theory
2016-07-20 v1 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.1212.1389,
title = {A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials},
author = {S. Ali Altug and Sandro Bettin and Ian Petrow and Rishikesh and Ian Whitehead},
journal= {arXiv preprint arXiv:1212.1389},
year = {2016}
}