We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.

This paper introduces the framework of decomposable combinatorial structures and their traversal algorithms. A combinatorial type is decomposable if it admits a specification in terms of unions, products, sequences, sets, and cycles, either in the labelled or in the unlabelled context. Many properties of decomposable structures are decidable. Generating function equations, counting sequences, and random generation algorithms can be compiled from specifications. Asymptotic properties can be determined automatically for a reasonably large subclass. Maple libraries that implement such decision procedures are briefly surveyed (LUO, combstruct, equivalent). In addition, libraries for manipulating holonomic sequences and functions are presented (gfun, Mgfun).

We describe a rational algorithm that computes the full partial fraction expansion of a rational function over the algebraic closure of its field of definition. The algorithm uses only gcd operations over the initial field but the resulting decomposition is expressed with linear denominators. We give examples from its Axiom and Maple implementations. Introduction The partial fraction decomposition of a rational function is a form where both the local and global behaviour of the function are easy to find. This is used when computing a primitive by hand, or any linear operation which is most easily done on a pole. An example is the efficient computation of asymptotic expansion of the solutions of a linear recurrence with constant coefficients [4]. Let f = A=D be a rational function in some field K(z). By the fundamental theorem of algebra, it is clear that f admits a partial fraction decomposition of the form f = P + X D(ff)=0 n ff X i=1 b ff;i (z \Gamma ff) i ; (1) where P is...

Computer algebra systems can be of help in the asymptotic analysis of combinatorial sequences. Several algorithms are presented, most of which have been implemented in Maple.