Higher-Order Linear Differential Equations for Unitary Matrix Integrals: Applications and Generalisations
Peter J. Forrester, Fei Wei
TL;DR
The paper develops a unified linear-differential framework for unitary matrix integrals with determinant insertions, proving that a first-order $(l+1)\times(l+1)$ matrix differential equation and a scalar degree $l+1$ differential equation govern $\langle (\det U)^q e^{s^{1/2} \operatorname{Tr}(U+U^\dagger)}\rangle_{U(l)}$ for general $q$ and $\beta$. This approach yields efficient power-series computation and connects to both combinatorial LIS problems ($q=0$) and zeta-function derivative moments ($q=l$), while offering a beta-generalisation framework and a Jack-polynomial hypergeometric function representation that underpins the differential-equation structure. The matrix-differential formulation leads to explicit recurrence relations for the power-series coefficients and scalable recurrences for $T_l(N)$, and it recovers, complements, or generalises existing Painlevé-based characterisations. By providing explicit higher-$l$ scalars and comparing with $\sigma$-Painlevé III$'$ forms, the work offers a robust, computable alternative to nonlinear approaches and deepens connections between random matrix theory, combinatorics, and number theory. Overall, the results yield practical algorithms for enumerative and number-theoretic questions and unify several analytic representations of unitary matrix integrals under a linear-differential umbrella.
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 $σ$-Painlev&é III$'$ second-order nonlinear differential equation. We show too that the natural $β$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.