Nested sums of symbols and renormalised multiple zeta functions
Nested sums of symbols and renormalised multiple zeta functions
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符号的嵌套和以及重整化的多个 zeta 函数
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
S. Paycha
中科院分区：
文献类型：
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作者：
D. Manchon;S. Paycha
我们定义了实线上符号在整数点上的离散嵌套和，当它们收敛时服从填充关系。它们通过欧拉-麦克劳林公式与符号的陈积分有关。利用适当的全纯正则化和Birkhoff分解，我们定义了满足填充关系的符号的重整化嵌套和。对于适当的符号，它们会产生重归一化的多个zeta函数，这些函数在所有参数上都满足填充关系。Hurwitz多重zeta函数也适用于这个框架。我们证明了在非正整数参数下多个zeta值的合理性，并研究了一个高维的模拟。
We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic regularisation followed by a Birkhoff factorisation, we define renormalised nested sums of symbols which also satisfy stuffle relations. For appropriate symbols they give rise to renormalised multiple zeta functions which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta functions fit into the framework as well. We show the rationality of multiple zeta values at nonpositive integer arguments, and a higher-dimensional analog is also investigated.