18 research outputs found
Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams
Nested sums containing binomial coefficients occur in the computation of
massive operator matrix elements. Their associated iterated integrals lead to
alphabets including radicals, for which we determined a suitable basis. We
discuss algorithms for converting between sum and integral representations,
mainly relying on the Mellin transform. To aid the conversion we worked out
dedicated rewrite rules, based on which also some general patterns emerging in
the process can be obtained.Comment: 13 pages LATEX, one style file, Proceedings of Loops and Legs in
Quantum Field Theory -- LL2014,27 April 2014 -- 02 May 2014 Weimar, German
Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
We propose a new method to compute the unipotent radical of the
differential Galois group associated to a parameterized second-order
homogeneous linear differential equation of the form
where is a rational
function in with coefficients in a -field of characteristic zero,
and is a commuting set of parametric derivations. The procedure developed
by Dreyfus reduces the computation of to solving a creative
telescoping problem, whose effective solution requires the assumption that the
maximal reductive quotient is a -constant linear differential
algebraic group. When this condition is not satisfied, we compute a new set of
parametric derivations such that the associated differential Galois
group has the property that is -constant, and such
that is defined by the same differential equations as . Thus
the computation of is reduced to the effective computation of
. We expect that an elaboration of this method will be successful in
extending the applicability of some recent algorithms developed by Minchenko,
Ovchinnikov, and Singer to compute unipotent radicals for higher order
equations.Comment: 12 page
Reduction-Based Creative Telescoping for Algebraic Functions
Continuing a series of articles in the past few years on creative telescoping
using reductions, we develop a new algorithm to construct minimal telescopers
for algebraic functions. This algorithm is based on Trager's Hermite reduction
and on polynomial reduction, which was originally designed for hyperexponential
functions and extended to the algebraic case in this paper
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page