5 research outputs found
Standard Relations of Multiple Polylogarithm Values at Roots of Unity
Let be a positive integer. In this paper we shall study the special
values of multiple polylogarithms at th roots of unity, called multiple
polylogarithm values (MPVs) of level . These objects are generalizations of
multiple zeta values and alternating Euler sums, which was studied by Euler,
and more recently, many mathematicians and theoretical physicists.. Our primary
goal in this paper is to investigate the relations among the MPVs of the same
weight and level by using the regularized double shuffle relations, regularized
distribution relations, lifted versions of such relations from lower weights,
and seeded relations which are produced by relations of weight one MPVs. We
call relations from the above four families \emph{standard}. Let be
the \Q-dimension of \Q-span of all MPVs of weight and level . Then
we obtain upper bound for by the standard relations which in general
are no worse or no better than the one given by Deligne and Goncharov depending
on whether is a prime-power or not, respectively, except for 2- and
3-powers, in which case standard relations seem to be often incomplete whereas
Deligne shows that their bound should be sharp by a variant of Grothedieck's
period conjecture. This suggests that in general there should be other linear
relations among MPVs besides the standard relations, some of which are written
down in this paper explicitly with good numerical verification. We also provide
a few conjectures which are supported by our computational evidence.Comment: By numerical computation we propose more non-standard relation
Reducibility of Signed Cyclic Sums of Mordell-Tornheim Zeta and L-Values
In this paper, we shall show that certain signed cyclic sums of
Mordell-Tornheim L-values are rational linear combinations of products of
multiple L-values of lower depths (i.e., reducible). This simultaneously
generalizes some results of Subbarao and Sitaramachandrarao, and Matsumoto et
al. As a direct corollary, we can prove that for any integer k>2 and positive
integer n, the Mordell-Tornheim sums zeta_\MT(\{n\}_k) is reducible
A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
In this work we present the computer algebra package HarmonicSums and its
theoretical background for the manipulation of harmonic sums and some related
quantities as for example Euler-Zagier sums and harmonic polylogarithms.
Harmonic sums and generalized harmonic sums emerge as special cases of
so-called d'Alembertian solutions of recurrence relations. We show that
harmonic sums form a quasi-shuffle algebra and describe a method how we can
find algebraically independent harmonic sums. In addition, we define a
differentiation on harmonic sums via an extended version of the Mellin
transform. Along with that, new relations between harmonic sums will arise.
Furthermore, we present an algorithm which rewrites certain types of nested
sums into expressions in terms of harmonic sums. We illustrate by nontrivial
examples how these algorithms in cooperation with the summation package Sigma
support the evaluation of Feynman integrals
Dirichlet type extensions of Euler sums
In this paper, we study the alternating Euler -sums and -sums, which
are infinite series involving (alternating) odd harmonic numbers, and have
similar forms and close relations to the Dirichlet beta functions. By using the
method of residue computations, we establish the explicit formulas for the
(alternating) linear and quadratic Euler -sums and -sums, from which,
the parity theorems of Hoffman's double and triple -values and
Kaneko-Tsumura's double and triple -values are further obtained. As
supplements, we also show that the linear -sums and -sums are
expressible in terms of colored multiple zeta values. Some interesting
consequences and illustrative examples are presented