Frédéric Chyzak

An extension of Zeilberger's fast algorithm to general holonomic functions

We extend Zeilbergers fast algorithm for definite hypergeometric summation to non-hypergeometric holonomic sequences. The algorithm generalizes to the differential case and to q-calculus as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.

Effective algorithms for parametrizing linear control systems over Ore algebras

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), multidimensional discrete systems, multidimensional convolutional codes, etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or π-free.

OreModules: A Symbolic Package for the Study of Multidimensional Linear Systems

In the seventies, the study of transfer matrices of time-invariant linear systems of ordinary differential equations (ODEs) led to the development of the polynomial approach [20, 22, 44]. In particular, the univariate polynomial matrices play a central role in this approach (e.g., Hermite, Smith and Popov forms, invariant factors, primeness, Bezout/Diophantine equations).

Differential equations for algebraic functions

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 exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series.

Hermite reduction and creative telescoping for hyperexponential functions

We present a new reduction algorithm that simultaneously extends Hermites 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 for bivariate hyperexponential functions. One of its main features is that it can avoid the costly computation of certificates. Its implementation outperforms Maples function DEtools[Zeilberger]. We also derive an order bound on minimal telescopers that is tighter than the known ones.

Complexity of creative telescoping for bivariate rational functions

The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilbergers 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 to bivariate rational functions. By considering this constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction. We then use our new method to compute diagonals of rational power series arising from combinatorics.

The Construction of Orthonormal Wavelets Using Symbolic Methods and a Matrix Analytical Approach for Wavelets on the Interval

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 perform this task. Moreover, we present a matrix analytical approach that unifies constructions of wavelets on the interval.

A non-holonomic systems approach to special function identities

We extend Zeilbergers 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 dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before.

The distribution of patterns in random trees

Let 𝒯n denote the set of unrooted labelled trees of size n and let m be a particular (finite, unlabelled) 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 is asymptotically normal with mean value and variance asymptotically equivalent to μn and σ2n, respectively, where the constants μ>0 and σ≥0 are computable.

On the existence of telescopers for mixed hypergeometric terms

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on additive and multiplicative decompositions. The criterion enables us to determine the termination of Zeilbergers algorithms for mixed hypergeometric inputs, and to verify that certain indefinite sums do not satisfy any polynomial differential equation.