Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 015, 21 pages      arXiv:2508.20797      https://doi.org/10.3842/SIGMA.2026.015

Higher-Order Linear Differential Equations for Unitary Matrix Integrals: Applications and Generalisations (with an Appendix by Folkmar Bornemann)

Peter J. Forrester a and Fei Wei b
a) School of Mathematics and Statistics, The University of Melbourne,Victoria 3010, Australia
b) Department of Mathematics, University of Sussex, Brighton, BN1 9RH, UK

Received September 04, 2025, in final form January 26, 2026; Published online February 18, 2026

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 $\sigma$-Painlevé 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.

Key words: unitary matrix integral; matrix linear differential equation; random permutation; Riemann zeta function; Painlevé equation.

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