Algorithmes efficaces pour le télescopage créatif utilisant des réductions

Creative telescoping is an algorithmic methodintroduced by Zeilberger for computing linear functionalequations satisfied by parametric sums and integrals.Initially developed for sums of hypergeometric functionsand integrals of hyperexponential functions, the methodhas since been extended to broader classes of functions,including ∂-finite and holonomic functions. In recentyears, a new family of creative telescoping algorithmsbased on reductions has emerged, addressing efficiencylimitations in earlier approaches. This thesis continues inthis direction by introducing two new reduction-basedalgorithms. The first one is designed for computingunivariate sums of ∂-finite functions, and the second onefor computing multivariate integrals of holonomicfunctions.Creative telescoping algorithms rely on the knowledge ofsufficiently many linear functional equations satisfied bythe function f to be summed or integrated. Theseequations are represented by a left ideal in an algebra ofoperators, known as the annihilator of f . For ∂-finitefunctions, this ideal resides in an Ore algebra, while forholonomic functions, it lies within a Weyl algebra. In the∂-finite case, annihilators are typically easy to compute.In contrast, obtaining a complete set of annihilatingoperators for holonomic functions is significantly morechallenging. When a function is both ∂-finite andholonomic, one approach for computing its holonomicannihilator consists in computing its ∂-finite annihilatorand performing an operator called the Weyl closure.Although an algorithm for computing the Weyl closureexists, it suffers from practical inefficiency. This thesisproposes a new algorithm for computing holonomicapproximations of the Weyl closure of left ideals in theWeyl algebra.