Efficient accelero-summation of holonomic functions

Refined Holonomic Summation Algorithms in Particle Physics

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle indefinite nested sums and products in the setting of difference rings and holonomic sequences. Relevant mathematics is reviewed and the underlying advanced difference ring machinery is elaborated upon. The flexibility of this new toolbox contributed substantially to evaluating complicated...

Creative Telescoping for Holonomic Functions

The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection o...

B-functions and Holonomic Systems

for some differential operator P(s, x, Dx) which is a polynomial on s. Professor M.Sato introduced the notions of "a-function", "b-function" and "'c-function" for relative invariants on prehomogeneous vector spaces, when he studied the fourier transforms and ~-functions associated with them (see [10, 12]). He defined, in the same time, b-functions for arbitrary holomorphic functions and conject...

Uniformly fast evaluation of holonomic functions

G-functions and Multisum versus Holonomic Sequences

A holonomic (i.e., D-finite, or P -recursive) sequence is one that satisfies a linear recursion relation with polynomial coefficients. A multisum sequence is one that is given by a multisum of a proper hypergeometric term. A fundamental theorem of Wilf-Zeilberger states that every multisum sequence is holonomic. For over 15 years, it was accepted as a reasonable conjecture that the converse hol...