标题： 高阶线性微分方程用于酉矩阵积分：应用与推广 显示英文标题

标题： Higher order linear differential equations for unitary matrix integrals: applications and generalisations

摘要： 我们考虑单位矩阵积分类$ \langle (\det U)^q e^{s^{1/2} {\rm Tr}(U + U^\dagger)} \rangle_{U(l)}$的特征，通过一个大小为$l+1$的向量函数的一阶矩阵线性微分方程，以及通过一个次数为$l+1$的标量线性微分方程。将展示后者是从前者得出的。矩阵线性微分方程提供了一种有效计算矩阵积分幂级数展开的方法，其中与$q=0$和$q=l$相关的矩阵积分分别与随机排列中最长递增子序列的计数以及临界线上黎曼zeta函数一阶和二阶导数的矩的问题有关。该方法与从涉及$\sigma$-Painlevé III$'$二阶非线性微分方程的已知特征中得出的方法进行了比较。 我们也证明了单位群积分的自然 $\beta$ 推广可以通过同一类线性微分方程来表征。 显示英文摘要

摘要： We consider characterisations of the class of unitary matrix integrals $ \langle (\det U)^q e^{s^{1/2} {\rm Tr}(U + U^\dagger)} \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 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 that the natural $\beta$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.