A recursion formula for moments of derivatives of random matrix polynomials
A recursion formula for moments of derivatives of random matrix polynomials
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随机矩阵多项式导数矩的递推公式
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发表时间：
2012
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通讯作者：
Ian Whitehead
中科院分区：
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作者：
S. A. Altug;S. Bettin;Ian Petrow;Rishikesh;Ian Whitehead
给出了USp（2N）、SO（2N）和O^-(2N)群上特征多项式导数的随机矩阵平均的渐近公式。这些平均值用于预测数论中出现的l函数导数矩的渐近公式。每个公式用超几何函数的行列式给出渐近的前导常数。我们发现了这些行列式之间的微分递归关系，它允许用k-th和(k-1)-st快速计算(k+1)-st常数。这种递归式使人联想到与painleve微分方程相关的au函数理论中出现的Toda格方程。
We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of au-functions associated with Painlev'e differential equations.