Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

TL;DR

This paper introduces classes of nested sums, iterated integrals, and special constants relevant to Feynman diagram evaluations, highlighting their algebraic structures, generalizations, and analytic continuations.

Contribution

It provides a comprehensive overview of the algebraic and structural properties of nested sums and their generalizations used in quantum field theory calculations.

Findings

Identification of algebraic relations via shuffle and stuffle algebras.

Extension of harmonic sums to generalized and cyclotomic sums.

Discussion of analytic continuation of nested sums to complex values.

Abstract

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-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound N.