多重zeta值倒数的有理逼近与三变量Cauchy数
Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers
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中文总结 AI 辅助
本文通过多重对数倒数的Laurent展开定义三变量Cauchy数，证明高阶Gregory系数有零点且序列最终交错，并猜想及部分证实其最终正性，同时给出zeta值倒数的新恒等式族。
中文摘要 AI 辅助
本文通过多重对数函数任意正整数次幂（称为阶）的倒数的Laurent展开，研究第一类Cauchy数（也称为Gregory系数）和第二类Cauchy数（也称为Nörlund数）的三变量推广。在对数情形下，我们将用WZ方法证明：对每个阶$\ell>1$，某个$\ell$阶Gregory系数必定为零，这与所有经典Gregory系数均非零的事实形成对比。我们还在该高阶对数情形下证明：对每个固定阶，序列最终是交错符号的，这一性质为经典Gregory系数所享有。在最一般的设定中，我们猜想这些新序列最终均为正，该猜想得到强有力的数值证据支持。最后，我们在多重对数函数和双重对数函数的特殊情形下证实了这一猜想。作为副产品，对每个zeta值和双重zeta值，我们找到无穷多个恒等式族，将其倒数表示为一个有理数与一个反常积分之和。
英文摘要
In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called Nörlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.
发表机构
- School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）
- Department of Mathematics, The Bishop’s School（主教中学数学系）
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