The gfun package provides tools for determining and manipulating generating functions.

You can perform computations with generating functions defined by equations. For example, given two generating functions defined by linear differential equations with polynomial coefficients, there is a procedure to compute the differential equation satisfied by their product.

Each command in the gfun package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, gfun:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands.

The following is a list of available commands for differential equations and recurrences.

There are different types of generating functions that you can manipulate using the gfun package, for example, ordinary (ogf) and exponential (egf) generating functions. For more information on the predefined generating function types, see gftypes.

Given the first terms of the sequence, the gfun package also contains functions that help determine generating functions.  In some cases, this answer is explicit. However, in most cases, an explicit expression does not exist, and the answer is an equation (either algebraic or differential) satisfied by the generating function.

The indexed names _C[0], _C[1],... are used by gfun to represent arbitrary constants. If such a name is given in the input, it might not be preserved during the computation.

The following is a list of commands available for numbers and series.

You can obtain information about computations by setting infolevel[gfun] to 1 through 5.

To display the help page for a particular gfun command, see Getting Help with a Command in a Package.

Salvy, B., and Zimmermann, P. "GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable". ACM Transactions on Mathematical Software. Vol. 20 No. 2. (1994): 163-177.