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

J Ablinger, J Blümlein, C Schneider

TL;DR

The paper surveys mathematical structures arising in multi-loop Feynman diagrams, focusing on multiply nested sums (harmonic sums) and their inverse Mellin transforms into iterated integrals (harmonic polylogarithms), and extends these frameworks to cyclotomic, generalized, and binomial-weighted variants, as well as to elliptic integrals. It highlights how Mellin transforms connect sums to iterated integrals and how special numbers such as multiple zeta values emerge as limits, while establishing algebraic (shuffle and stuffle) and structural relations that enable basis reduction. The work also documents computational tools (Sigma, HarmonicSums) and analytic continuation methods to complex values of the Mellin variable $N$, enabling precise high-loop calculations such as Wilson coefficients and operator matrix elements. Together, these contributions illustrate a hierarchical, interrelated landscape of functions and constants that expand the analytic toolkit for perturbative quantum field theory and point toward richer structures at higher loops and legs.

Abstract

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.

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

TL;DR

The paper surveys mathematical structures arising in multi-loop Feynman diagrams, focusing on multiply nested sums (harmonic sums) and their inverse Mellin transforms into iterated integrals (harmonic polylogarithms), and extends these frameworks to cyclotomic, generalized, and binomial-weighted variants, as well as to elliptic integrals. It highlights how Mellin transforms connect sums to iterated integrals and how special numbers such as multiple zeta values emerge as limits, while establishing algebraic (shuffle and stuffle) and structural relations that enable basis reduction. The work also documents computational tools (Sigma, HarmonicSums) and analytic continuation methods to complex values of the Mellin variable , enabling precise high-loop calculations such as Wilson coefficients and operator matrix elements. Together, these contributions illustrate a hierarchical, interrelated landscape of functions and constants that expand the analytic toolkit for perturbative quantum field theory and point toward richer structures at higher loops and legs.

Abstract

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.

Paper Structure

This paper contains 8 sections, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Harmonic Sums, Polylogarithms and Multiple Zeta Values
  3. Cyclotomic Harmonic Sums, Polylogarithms, and Numbers
  4. Generalized Harmonic Sums, Polylogarithms, and Numbers
  5. Nested Binomial Sums
  6. Elliptic Integrals
  7. Analytic Continuations the various Sums
  8. Conclusions

Figures (1)

  • Figure 1: Connection between harmonic sums (H-Sums), S-sums (S-Sums) and cyclotomic harmonic sums (C-Sums), their values at infinity and harmonic polylogarithms (H-Logs), generalized harmonic polylogarithms (G-Logs) and cyclotomic harmonic polylogarithms (C-Logs) and their values at special constants.