Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials

TL;DR

This paper develops and analyzes cyclotomic harmonic sums and polylogarithms, extending their algebraic and structural understanding for applications in quantum field theory calculations involving complex nested sums.

Contribution

It introduces cyclotomic harmonic polylogarithms and sums, deriving their algebraic relations, analytic continuations, and basis representations up to weight 2 and cyclotomy 20.

Findings

Derived algebraic and structural relations for cyclotomic harmonic sums.

Performed analytic continuation to complex N for these sums.

Established basis representations for sums up to weight 2 and cyclotomy 20.

Abstract

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 Poincar\'e--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}.