在可重正化量子场论的大规模高阶微扰计算中，Feynman积分的计算需要多重套调和和的延拓，而多重套调和和可以通过Poincaré迭代积分的Mellin变换（包括高阶分圆多项式的展开子）生成为实表示.我们导出了分圆调和多项式与调和和，并研究了它们的代数关系和结构关系。利用解析表示法将分圆调和和解析延拓到$N$的复值.我们还考虑了分圆调和多项式在自变量$x=1$上的特殊值，对于与色重zeta值有关的$N \rightarrow \infty$上的分圆调和和，基于stuffle代数和shuffle代数以及三个重宗量关系，导出了它们之间的各种关系.我们还考虑了与无限分圆调和和相关的单位根上的无限广义套调和和。导出了权重{ \sf w = 1，2}和到分圆{ \sf l = 20}的基表示。
英文摘要：
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré--iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of $N$ is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument $x=1$, resp., for the cyclotomic harmonic sums at $N \rightarrow \infty$, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight {\sf w = 1,2} sums up to cyclotomy {\sf l = 20}.
Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6
本文导出了权为{ \sf w = 6}的Nielsen积分和调和多项式的套调和和与相应的Mellin变换之间的结构关系。它们出现在QED和QCD中无质量单尺度量的计算中，如反常尺寸和Wilson系数，到3-和4-圈级。我们考虑权为6的多重调和和的集合，而没有指标$\{-1\}$。这个限制对于所有已知的物理情况是足够的。结构关系补充代数关系，由于调和和之间的洗牌产品，研究较早。在权重{ \sf w = 6}处贡献的486个可能谐波和的原始量减少到99个没有索引$\{-1\}$的和。代数和结构关系导致进一步减少到20个基本功能。这些函数补充了15个基本函数的集合，直到前面导出的权重{ \sf w = 5}。给出了求复平面上基本和的解析表示的算法。
The $q$-extension of iterated integrals and nested sums in quantum field theory
Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Institut für Theoretische Physik III, IV, TU Dortmund（多特蒙德工业大学理论物理第三、四研究所）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）；Johannes Kepler University Linz, Research Institute for Symbolic Computation (RISC)（林茨约翰内斯·开普勒大学符号计算研究所）
The $μ$-extension of iterated integrals and nested sums
Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Institut für Theoretische Physik III, IV, TU Dortmund（多特蒙德工业大学理论物理第三、四研究所）；Bogolyubov Institute for Theoretical Physics（博戈柳博夫理论物理研究所）
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
