We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular K Bessel functions. Earlier studies have identified the same determinant, but with the K Bessel functions replaced by I Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a τ-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III′, and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same σ-form Painlevé III′ differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the n →∞ limit, both with the Laguerre parameter α fixed, and with α = n (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

中文翻译：

---

来自冈本哈密顿公式的 PIII' τ 函数序列在随机矩阵理论中的应用

我们考虑由特征值倒数之和组成的拉盖尔酉系综 (LUE) 的奇异线性统计量。可以观察到，该统计量的指数生成函数可以写成 Toeplitz 行列式，其中条目根据特定ķ贝塞尔函数。早期的研究已经确定了相同的决定因素，但与ķ贝塞尔函数替换为一世在非负整数拉盖尔参数的情况下，贝塞尔函数与 LUE 的广义间隙概率的硬边缘缩放限制有关。我们表明，由这两个贝塞尔函数的任意线性组合形成的 Toeplitz 行列式作为τ-Okamoto 的 Painlevé III 哈密顿公式中的函数序列'，因此两个 Toeplitz 行列式的对数导数满足相同σ-form Painlevé III'微分方程，给出了一个可以从早期结果中观察到的事实的解释。此外，对生成函数的这种表征与其在n →∞限制，都带有拉盖尔参数α固定，并与α = n（后一种情况与对 Wigner 时间延迟统计量的分布的应用相关）给出。