量子场论中迭代积分与嵌套和的q-扩展
The $q$-extension of iterated integrals and nested sums in quantum field theory
AI总结：
本文构造量子场论中迭代积分与嵌套和的q-扩展特殊函数，推导其微分、差分方程及闭式解，该类函数可用于q-变形对易关系下的微扰计算。
AI中文摘要：
微扰量子场论中零标度与单标度量的解析计算会得到特殊数与特殊函数，过去数十年间已揭示了其中首批结果，这些结果是多对数函数的推广，形式为针对特殊字母表的库默尔-庞加莱迭代积分及其扩展。随着耦合常数阶数的增长，多对数函数、尼尔森积分、针对线性分母项的迭代积分、分圆字母、二次型诱导的字母、平方根值字母以及更一般的函数均有贡献。对于嵌套和，我们考虑嵌套调和和、广义调和和、由二次型导出的嵌套和、分圆调和和以及包含中心二项式的嵌套和。我们构造这些特殊函数及与之关联的嵌套和的q-扩展，它们与x=0处的级数展开及无q情形下的梅林变换相关。这些函数有望在q-变形对易关系情形下的微扰计算中发挥作用。对于较简单的函数空间，给出了闭式解；对于更复杂的字母表，给出了针对各类情形推导q-扩展的算法步骤，还推导了这些高等超越函数的决定性微分方程与差分方程。q-扩展的特殊函数与对应的μ-扩展函数差异显著。
英文摘要：
Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $μ$-extended functions.
