87 research outputs found
Iterated Binomial Sums and their Associated Iterated Integrals
We consider finite iterated generalized harmonic sums weighted by the
binomial 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 and the iterated integrals at 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 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 . 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.Comment: 62 pages Latex, 1 style fil
How to generate all possible rational Wilf-Zeilberger pairs?
A Wilf--Zeilberger pair in the discrete case satisfies the equation
. We present a structural
description of all possible rational Wilf--Zeilberger pairs and their
continuous and mixed analogues.Comment: 17 pages, add the notion of pseudo residues in the differential case,
and some related papers in the reference, ACMES special volume in the Fields
Institute Communications series, 201
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