Introduction to the gfun Package

Description

• The gfun package has been designed as a help for the manipulation and discovery of functions or sequences satisfying linear differential or recurrence equations. The name of the package comes from its combinatorial application to generating functions.
• The basic principle of the package is that linear differential equations or recurrences can be used as data-structures to represent their solutions. Procedures that convert to such a representation are: algfuntoalgeq algeqtodiffeq holexprtodiffeq
• The differential equations and recurrences can then be manipulated by algebraicsubs diffeqtorec rectodiffeq poltodiffeq poltorec reducerecorder
• and the following functions that perform more elementary operations borel cauchyproduct diffeq+diffeq diffeq*diffeq diffeqtohomdiffeq hadamardproduct invborel Laplace rec+rec rec*rec rectohomrec minimizediffeq (new) istranscendental (new) indicialpolynomial (new)
• Useful information can then be extracted from these equations by one of algeqtoseries ratpolytocoeff rectoproc the NumGfun subpackage the ContFrac subpackage
• Given the first terms of the sequence, the gfun package also contains functions that will help conjecture what the generating function is. In some cases, this answer will be ``explicit''. In most cases though, such an explicit expression will not exist, and the answer will be an equation (either differential or algebraic) satisfied by the generating function. The functions dealing with numbers and series are: listtodiffeq seriestodiffeq listtorec seriestorec listtoalgeq seriestoalgeq listtoratpoly seriestoratpoly listtohypergeom seriestohypergeom guesseqn guessode (new) guessgf guessrelation (new) hermite_pade (new)

The way the guessing proceeds can be controled by modifying Parameters.

There are different types of generating functions that can be manipulated in gfun, such as ordinary (ogf) and exponetial (egf) generating functions. For a description of the predefined types see gftypes.

The following conversion routines transform from one type to another:

listtolist listtoseries seriestolist seriestoseries

• Information about the computations that are being done can be obtained by setting infolevel[gfun] to anything between 1 and 5.
• Coefficients and truncated series can be extracted directly from differential equations and recurrences with diffeqtolist, diffeqtoseries, rectolist, and rectoseries.
• For relations between several series, use guessrelation when a candidate should be validated against held-out terms, hermite_pade when all supplied terms should be used for a Hermite-Pade approximant, or approximantbasis when the complete modular relation basis is wanted.
• The specialized listto* and seriesto* commands are the usual entry points when the desired result type is known. These commands share relation-finding algorithms based mainly on Hermite-Pade approximation and matrix rational interpolation.

References

More information (on an early version) can be found in: "Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable.", B. Salvy and P. Zimmermann, ACM Transactions on Mathematical Software, 20 (2), 163--177 (1994).

See also the web page of gfun: http://perso.ens-lyon.fr/bruno.salvy/software/the-gfun-package/

See Also

with gfun[Parameters]