Harmonic analysis of iterated function systems with overlap

Iterated Binomial Sums and their Associated Iterated Integrals

We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for $N \rightarrow \infty$ and the iterated integrals at $x=1$ lea…

Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers

A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of special numbers. Starting with harmonic sums and polylogarithms we discuss recent extensions of these quantities as cyclotomic, generalized (cyclotomic), and binomially weighted sums, associated iterated integrals and special constants and their relations.

General spherical harmonic bra-ket overlap integrals of trigonometric functions

Closed formulas in terms of double sums of Clebsch-Gordan coefficients are computed for the evaluation of bra-ket spherical harmonic overlap integrals of a wide class of trigonometric functions. These analytical expressions can find useful application in problems involving non-separable wave equations, e.g. general-relativistic perturbation theory, electromagnetism, quantum theory, etc, wherein the overlap integrals arise from the coupling among different angular modes.

Computer Algebra Algorithms for Special Functions in Particle Physics

This work deals with special nested objects arising in massive higher order perturbative calculations in renormalizable quantum field theories. On the one hand we work with nested sums such as harmonic sums and their generalizations (S-sums, cyclotomic harmonic sums, cyclotomic S-sums) and on the other hand we treat iterated integrals of the Poincaré and Chen-type, such as harmonic polylogarithms and their generalizations (multiple polylogarithms, cyclotomic harmonic polylogarithms). The iterated integrals are connected to the nested sums via (generalizations of) the Mellin-transformation and …

Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomia…

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

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。