Non-commutative elimination in Ore algebras proves multivariate identities
type: Item
instance of: scholarly article
title: Non-commutative elimination in Ore algebras proves multivariate identities
zbMATH Open document ID: 0944.05006
author: Frédéric Chyzak
author: Bruno Salvy
published in: Journal of Symbolic Computation
publication date: 1999-01-24T00:00:00Z
full work available at URL: holonomy.pdf
review text: Many special functions are solutions of holonomic systems. \textit{D. Zeilberger} [ J. Comput. Appl. Math. 32, No. 3, 312-368 (1990; Zbl 0738.33001)] showed that many special function identities can be proved in a finite number of steps by invoking J. Bernstein's theory of holonomic systems. In this paper the authors generalize the notions of \(P\)-finite functions and sequences used by Zeilberger and develop a theory of \(\partial\)-finite functions and sequences. This theory makes it possible to use on a general level skew polynomial algebras as the algebraic setting and non-commutative Gröbner bases as a tool to develop effective algorithms. The paper provides a very general framework for computing sums and integrals and for the proof and the discovery of multivariate identities. The authors illustrate the theory by well-chosen examples making use of an interesting Maple package (written by the first author).
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 33F10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1204263
zbMATH Keywords: combinatorial identities
zbMATH Keywords: special function identities
zbMATH Keywords: Ore algebras
zbMATH Keywords: holonomic systems
zbMATH Keywords: functions and sequences
reviewed by: Friedrich Schwarz
describes a project that uses: Maple
describes a project that uses: gfun
describes a project that uses: qZeil
describes a project that uses: Mgfun
describes a project that uses: pqTelescope
MaRDI profile type: Publication
OpenAlex ID: W1977049875
DOI: 10.1006/JSCO.1998.0207
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description: scientific article; zbMATH DE number 1204263