广义调和和与多项式的分析与数学性质
Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
摘要：
在最近的量子色动力学（QCD）中的大质量Feynman积分的三圈计算中，在最近的组合问题中，出现了所谓的广义调和和（简称$S$ -和）。它们的特征在于有理（或实）分子权重也不同于$\pm 1$。在这篇文章中，我们将系统地探讨这些和的算法和分析性质。我们工作的梅林和逆梅林变换连接的总和与相关的庞加莱迭代积分，也称为广义调和多项式。在这方面，我们得到明确的解析延拓的$S$ -和开始经常出现在当前的QCD计算的渐近展开的手段。此外，我们推导出代数和结构关系，如微分w.r.t.外和指标和不同的多变量关系，对$S$ -和式的紧化进行了研究.最后，我们计算代数关系的无限$S$ -总和，或等价的广义调和多项式在特殊值评估。相应的算法和关系被编码在计算机代数包{ \tt HarmonicSums}中。
英文摘要：
In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by rational (or real) numerator weights also different from $\pm 1$. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the $S$-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multi-argument relations, for the compactification of $S$-sum expressions. Finally, we calculate algebraic relations for infinite $S$-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package {\tt HarmonicSums}.
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