Iterated binomial sums and their associated iterated integrals
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 3loop 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 squareroot valued alphabets. The values of the sums for N → ∞ 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 → ∞ 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 {C}. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as, e.g., for multiscale processes. We also derive algorithms to transform iterated integrals over rootvalued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 November 2014
 DOI:
 10.1063/1.4900836
 arXiv:
 arXiv:1407.1822
 Bibcode:
 2014JMP....55k2301A
 Keywords:

 High Energy Physics  Theory;
 High Energy Physics  Phenomenology;
 Mathematical Physics
 EPrint:
 62 pages Latex, 1 style file