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 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)
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.
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)
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)
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)
Many combinatorial generating functions can be expressed as combinations of symmetric functions , or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential… (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)