On the Differential-Integral Analogue of Zeilberger's Algorithm to Rational Functions

Complexity of creative telescoping for bivariate rational functions

“…Essentially two algorithms for minimal telescopers can be found in the literature: The classical way [1] is to apply a , together with bounds on output differential analogue of Gosper's indefinite summation algorithm, which reduces the problem to solving an auxiliary linear differential equation for polynomial solutions. An algorithm developed later in [7] (see also [12]) performs Hermite reduction on f to get an additive decomposition of the form f = Dy(a) + m i=1 ui/vi, where the ui and vi are in k(x)[y] and the vi are squarefree. Then, the algorithm in [1] is applied to each ui/vi to get a telescoper Li minimal for it.…”

“…The least common left multiple of the Li's is then proved to be a minimal telescoper for f . This algorithm performs well only for specific inputs (both in practice and from the complexity viewpoint), but it inspired our Lemma 22 via [12].…”

“…As a first contribution in this article, we present a new, provably faster algorithm for computing minimal telescopers for bivariate rational functions. Instead of a single use of Hermite reduction as in [12], we apply Hermite reduction to the D i x (f )'s, iteratively for i = 0, 1, . .…”

Creative Telescoping for Parametrised Integration and Summation

“…As in the differential case, this means that Wilf and Zeilberger's approach and Zeilberger's fast algorithm both terminate on holonomic inputs. (Ore, 1929(Ore, , 1930Sato, 1990;Gel'fand, Graev, and Retakh, 1992) (Abramov, 2002(Abramov, , 2003Chen, Hou, and Mu, 2005) rational case: (Abramov and Le, 2000;Le, 2001;Abramov and Le, 2002) Wilf and Zeilberger's conjecture: (Abramov and Petkovšek, 2002;Hou, 2004) (voir aussi la thèse de Hou en 2001 (réferences dans (Abramov and Petkovšek, 2002), qui fait la conjecture dans le cas bivarié, en parallèle de AbPe (autres réfs)) rq : ici, holonome est défini comme P-récursif de Stanley ; il faudrait montrer/rappeler l'équivalence 2.1. Termination Criteria for Hypergeometric-Hyperexponential Terms.…”

On the existence of telescopers for mixed hypergeometric terms

“…However, the situation in other cases turns out to be more involved. In the discrete case, the first complete solution to the termination problem has been given by Le (2001) and Abramov and Le (2002), by deciding whether telescopers exist for a given bivariate rational sequence in the (q)-discrete variables y 1 and y 2 . According to their criterion, the rational sequence f = 1 y 2 1 + y 2 2 has no telescoper.…”