Fonctions holonomes en calcul formel

Cette these montre comment le calcul formel permet la manipulation d'une grande classe de suites et fonctions solutions d'operateurs lineaires, la classe des fonctions holonomes. Celle-ci contient de nombreuses fonctions speciales, en une ou plusieurs variables, et de nom- breuses suites de la combinatoire. Un cadre theorique est tout d'abord introduit pour algorith- miser les proprietes de cloture de la classe holonome, pour y permettre un test a zero et pour unifier les calculs differentiels sur les fonctions et les calculs de recurrences sur les suites. Ces methodes s'appuient sur des calculs par une extension de la theorie des bases de Grobner dans un cadre de polynomes non commutatifs, les polynomes de Ore. Deux types d'algorithmes de sommation et d'integration symboliques definies et indefinies sont ensuite developpes, dont la justification theorique fait appel a la theorie des D-modules holonomes. Les premiers ont recours a une elimination polynomiale non commutative par bases de Grobner ; les seconds a des algo- rithmes de resolution de systemes fonctionnels lineaires en leurs solutions fractions rationnelles. Bien plus que la recherche de formes closes, l'objectif est de pouvoir continuer a calculer avec la representation implicite des objets holonomes meme en l'absence de formes explicites. Ce type de calculs permet en particulier la preuve automatique d'identites sommatoires et integrales. Une implantation de ces algorithmes dans le systeme de calcul formel Maple a permis de donner la premiere preuve automatique d'identites jusqu'a present inaccessibles par le calcul formel.

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