Iterated Binomial Sums and their Associated Iterated Integrals

By Jakob Ablinger, J. Blumlein, Clemens G. Raab, Carsten Schneider. 2014

143 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/d30cc1b3bcaa062d597a2c370fbf6b557702cae0

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism (2017, background): “Our paper is not the first to consider elliptic generalisations of MPLs [14, 17, 19, 20, 24, 31, 45, 49, 50] or iterated integrals over integration kernels that involve square roots of polynomials of degree greater than two [51].”
Feynman integrals and hyperlogarithms (2015, methodology): “Recently significant progress in the evaluation of Feynman integrals was achieved in particular using modern summation techniques [3, 5, 131] and with considerable improvements of the method of differential equations [9, 91, 97, 99].”