# Higher-Order Linear Differential Equations for Unitary Matrix Integrals: Applications and Generalisations

## TL;DR

This paper develops higher-order linear differential equations to characterize unitary matrix integrals, enabling efficient computation and applications to permutation enumeration and Riemann zeta function moments.

## Contribution

It introduces a novel approach using higher-order linear differential equations for unitary matrix integrals, extending to beta ensembles and linking to combinatorial and number theory problems.

## Key findings

- Matrix differential equations efficiently compute matrix integral expansions.
- Connections established between matrix integrals and permutation enumeration.
- Application to moments of derivatives of the Riemann zeta function.

## Abstract

In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential equation for a vector function of size $l+1$, and in terms of a scalar linear differential equation of degree ${l+1}$. It will be shown that the latter follows from the former. The matrix linear differential equation provides an efficient way to compute the power series expansion of the matrix integrals, which with $q=0$ and $q=l$ are of relevance to the enumeration of longest increasing subsequences for random permutations, and to the question of the moments of the first and second derivative of the Riemann zeta function on the critical line, respectively. This procedure is compared against that following from known characterisations involving the $\sigma$-Painlev&\'e III$'$ second-order nonlinear differential equation. We show too that the natural $\beta$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.