Iterated Binomial Sums and their Associated Iterated Integrals

TL;DR

The paper develops a comprehensive, algorithmic framework linking finite nested binomial sums to Mellin transforms of iterated root-valued integrals. By introducing a root-valued alphabet and H^* iterated integrals, it provides systematic methods to obtain Mellin representations, perform convolutions, and transform between sums and integrals, enabling analytic continuation to complex N and facilitating asymptotic expansions. The work extends the landscape of special numbers beyond multiple zeta values and cyclotomic constants, with direct relevance to massive 3-loop Feynman diagrams and operator insertions. It also treats infinite (inverse) binomial sums via generating functions and presents multiple methods to compute Mellin transforms of D-finite functions, broadening the toolkit for high-precision loop calculations in quantum field theory.

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.

Theorems & Definitions (26)

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• Lemma 1
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