Frédéric Chyzak

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Many identities involving special functions, combinatorial sequences or their q-analogues can be proved using linear operators and simple arguments on the dimension of related vector spaces. In this article, we develop a theory of @-nite sequences and functions which provides a uniied framework to express algorithms proving and discovering mul-tivariate(More)
In the seventies, the wish to study transfer matrices of time-invariant finite-dimensional linear systems has led to the development of the polynomial approach in which the canonical forms of matrices of univariate polynomials were studied (e.g., Hermite and Smith forms, invariant factors, primeness). In the middle of the seventies, while generalizing(More)
We extend Zeilberger's fast algorithm for deenite hypergeometric sum-mation to non-hypergeometric holonomic sequences. The algorithm generalizes to diierential and q-cases as well. Its theoretical justiication is based on a description by linear operators and on the theory of holonomy. R esum e. Nous etendons l'algorithme rapide de Zeilberger pour la(More)
In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs),(More)
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there(More)
We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the(More)
We propose algorithms for the computation of the first <i>N</i> terms of a vector (or a full basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations that is quasi-linear with respect to <i>N</i>. Similar results are also given in the non-linear case. This extends previous(More)
The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilberger's framework of (differential) creative telescoping. Our complexity-driven approach is to obtain tight degree bounds on the various expressions involved in the method. To make the problem more tractable, we restrict(More)
We present a new reduction algorithm that simultaneously extends Hermite's reduction for rational functions and the Hermite-like reduction for hyperexponential functions. It yields a unique additive decomposition that allows to decide hyperexponential integrability. Based on this reduction algorithm, we design a new algorithm to compute minimal telescopers(More)