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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)

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, underde-termined systems of partial differential equations (PDEs),… (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)

In this paper we discuss closed form representations of filter coefficients of wavelets on the real line, half real line and on compact intervals. We show that computer algebra can be applied to achieve this task. Moreover , we present a matrix analytical approach that unifies constructions of wavelets on the interval.

Let Tn denote the set of unrooted labeled trees of size n and let M be a particular (finite, unlabeled) tree. Assuming that every tree of Tn is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of M as an induced subtree is asymptotically normal with mean value and variance asymptotically… (More)

We give a randomized algorithm in deterministic time O(N log M) for estimating the score vector of matches between a text string of length N and a pattern string of length M , i.e., the vector obtained when the pattern is slid along the text, and the number of matches is counted for each position. A direct application is approximate string matching. The… (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)

- Frédéric Chyzak
- 1998

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)