29 research outputs found
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
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
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 Definite Summation of D-finite Functions
Creative telescoping is an algorithmic method initiated by Zeilberger to
compute definite sums by synthesizing summands that telescope, called
certificates. We describe a creative telescoping algorithm that computes
telescopers for definite sums of D-finite functions as well as the associated
certificates in a compact form. The algorithm relies on a discrete analogue of
the generalized Hermite reduction, or equivalently, a generalization of the
Abramov-Petkov\v sek reduction. We provide a Maple implementation with good
timings on a variety of examples.Comment: 15 page