论文标题
在无限数量的非线性欧拉和
On an infinite number of nonlinear Euler sums
论文作者
论文摘要
在过去的几个世纪中，许多作者对线性谐波数量总和进行了研究，但是关于非线性Euler二次甚至更高程度的非线性Euler总和只有很少的结果。 Flajolet and Salvy在1997年发表了有关非线性EULER总和的首次系统研究，该研究已发表，随后是不同作者在过去几年中提出的类似研究。尽管这些研究仅限于提名人由偶数或奇数超谐波总和的产物组成的总和，而该分母为$ 1/k^n $。我们将这些结果概括为具有不同的分母和名称器的非线性Euler总和，这些总和还组成了偶数和奇数超声数之间的混合产物。详细说明，我们介绍了八个二次Euler总和的族，它们仅在非线性Euler总和始终是偶数数字的情况下，由Zeta值和特殊类型的线性Euler总和表达。我们发现的八个不同的非线性Euler总和由偶数和奇数超谐波数之间的各种产品组成，除以三种不同类型的分母$ 1/k^n $，$ 1/（（2K-1）^n）$和$ 1/（k（2k（2k-1）））$。计算方案基于适当的两值整数函数，这使我们能够以zeta值和奇数型线性谐波数，甚至超谐音数的第二顺序来明确计算这些序列。
Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums consisting of products of hyperharmonic sums had been published by Flajolet and Salvy in 1997 followed by similar studies presented during the last years by different authors. Although these studies had been restricted to sums where the nominator consists of a product of even or odd hyperharmonic sums, where the denominator is of the type $1/k^n$. We have generalized these results to nonlinear Euler sums with different denominators and nominators which consist in addition of mixed products between even and odd hyperharmonic numbers. In detail we present eight families of quadratic Euler sums which are expressible by zeta values and special types of linear Euler sums only where the order of the nonlinear Euler sums is always an even number. The resulting eight different families of nonlinear Euler sums which we discovered consist of various products between even and odd hyperharmonic numbers, divided by three different types of denominators $1/k^n$, $1/((2k-1)^n)$ and $1/(k(2k-1))$. The calculational scheme is based on proper two-valued integer functions, which allow us to compute these sequences explicitly in terms of zeta values and pairs of odd-type linear harmonic numbers and even hyperharmonic numbers of second order.
