TY  - JOUR
AU  - Ablinger, J.
AU  - Blümlein, J.
AU  - Raab, Clemens
AU  - Schneider, C.
TI  - Iterated Binomial Sums and their Associated Iterated Integrals
JO  - Journal of mathematical physics
VL  - 55
IS  - 11
SN  - 1089-7658
CY  - College Park, Md.
PB  - American Inst. of Physics
M1  - PUBDB-2015-01108
M1  - DESY-14-021
M1  - DO-TH-13/22
M1  - SFB/CPP-14-35
M1  - LPN 14-082
M1  - arXiv:1407.1822
SP  - 112301
PY  - 2014
N1  - OA
AB  - We consider finite iterated generalized harmonic sums weighted by the binomial ( \frac2kk ) 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 → ∞ 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  ∈ 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.
LB  - PUB:(DE-HGF)16
UR  - <Go to ISI:>//WOS:000345643100017
DO  - DOI:10.1063/1.4900836
UR  - https://bib-pubdb1.desy.de/record/207123
ER  -