双zeta函数欧拉约化公式的符号模拟
A signed analog of Euler's reduction formula for the double zeta function
摘要：
双zeta函数是由双狄利克雷级数定义的两个参数的函数，并在1742年由欧拉回应哥德巴赫的一封信中首次研究。通过计算许多例子，欧拉推导出了双zeta函数在两个自变量是具有相反奇偶性的正整数的情况下，根据黎曼ζ函数的值的封闭形式求值。在这里，我们考虑欧拉求值的一个有符号的模拟：即符号双zeta函数的一个约简公式，当符号专用为1时，该约简公式约简为欧拉求值。这个公式首先由Borwein、Bradley和Broadhurst在1997年的一篇论文中提出，随后由Flajolet和Salvy使用围道积分证明。本文的目的是给出一个基于部分分式恒等式的初等证明。
英文摘要：
The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formula was first stated in a 1997 paper by Borwein, Bradley and Broadhurst and was subsequently proved by Flajolet and Salvy using contour integration. The purpose here is to give an elementary proof based on a partial fraction identity.
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