Iterated Binomial Sums and their Associated Iterated Integrals

TL;DR

This paper explores binomially weighted iterated sums and integrals relevant in high-order Feynman diagram calculations, introducing new constants and algorithms for their systematic analysis and extension of known special number classes.

Contribution

It develops algorithms for Mellin representation, asymptotic expansion, and analytic continuation of binomial sums and integrals, extending the classes of special constants and functions used in quantum field theory.

Findings

New constants from sums at infinity and integrals at x=1.

Algorithms for Mellin transforms and asymptotic expansions.

Extended classes of special numbers beyond multiple zeta values.

Abstract

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 $x=1$ lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit $N \rightarrow \infty$ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to $N \in \mathbb{C}$. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as e.g. for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.