论文标题
特征多项式的联合力矩及其用于圆形$β$的衍生物
Joint moments of a characteristic polynomial and its derivative for the circular $β$-ensemble
论文作者
论文摘要
计算特征多项式关节矩的缩放限制以及特征多项式的衍生物的缩放限制的问题，该矩阵的矩阵具有HAAR测量的矩阵，首先是在与Riemann Zeta功能有关的研究中首次出现的。随后，Winn表明，这些关节矩可以等效地写作作为在Cauchy单一合奏中分布的痕迹的力矩，此外，基于Schur多项式的某些超测量函数，这启用了明确的计算。我们给出了这些结果的$β$总体化，现在插孔多项式扮演了schur多项式的作用。这会导致对所有$β> 0 $的缩放力矩的明确评估，但要受到以下约束，即其中的特定参数等于非负整数。还考虑了对jacobi $β$ endlemble的单数统计$ \ sum_ {j = 1}^n 1/x_j $的计算的计算。
The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble, and furthermore relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a $β$-generalisation of these results, where now the role of the Schur polynomials is played by the Jack polynomials. This leads to an explicit evaluation of the scaled moments for all $β> 0$, subject to the constraint that a particular parameter therein is equal to a non negative integer. Consideration is also given to the calculation of the moments of the singular statistic $\sum_{j=1}^N 1/x_j$ for the Jacobi $β$-ensemble.
