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@ARTICLE{Ablinger:170179,
      author       = {Ablinger, J. and Blümlein, J. and Raab, Clemens and
                      Schneider, C.},
      title        = {{I}terated {B}inomial {S}ums and their {A}ssociated
                      {I}terated {I}ntegrals},
      reportid     = {DESY-2014-03017, DESY-14-021. DO-TH-13/22. SFB/CPP-14-35.
                      LPN 14-082. arXiv:1407.1822},
      year         = {2014},
      note         = {62 pages Latex, 1 style file},
      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.},
      cin          = {ZEU-THEO},
      cid          = {I:(DE-H253)ZEU-THEO-20120731},
      pnm          = {514 - Theoretical Particle Physics (POF2-514)},
      pid          = {G:(DE-HGF)POF2-514},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)25 / PUB:(DE-HGF)15},
      eprint       = {1407.1822},
      howpublished = {arXiv:1407.1822},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:1407.1822;\%\%$},
      url          = {https://bib-pubdb1.desy.de/record/170179},
}