This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# GeneratingFunction

 GeneratingFunction gives the generating function in x for the sequence whose n series coefficient is given by the expression expr. GeneratingFunctiongives the multidimensional generating function in , , ... whose , , ... coefficient is given by expr.
• The generating function for a sequence whose n term is is given by .
• The multidimensional generating function is given by .
• The following options can be given:
 Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate answers that involve conditions on parameters Method Automatic method to use VerifyConvergence True whether to verify convergence
The generating function for the sequence whose n term is 1:
All coefficients in the series are 1:
Univariate generating function:
Multivariate:
The generating function for a shifted sequence:
The generating function for the sequence whose n term is 1:
 Out[1]=
All coefficients in the series are 1:
 Out[2]=

Univariate generating function:
 Out[1]=
Multivariate:
 Out[2]=

The generating function for a shifted sequence:
 Out[1]=
 Scope   (4)
Polynomials can be expressed in terms of rational functions:
Rational generating functions:
Generating function of a periodic sequence:
Special functions:
Compute the generating function at a point:
 Options   (5)
In general, this generating function cannot be given:
By providing additional Assumptions, a closed form can be given:
By default, no conditions are given for where a generating function is convergent:
Use GenerateConditions to generate conditions of validity:
Setting VerifyConvergence to False will treat generating functions as formal objects:
Setting VerifyConvergence to True will verify that the radius of convergence is nonzero:
In addition, setting GenerateConditions to True will display the conditions for convergence:
Use SeriesCoefficient to get the sequence from its generating function:
GeneratingFunction effectively computes an infinite sum:
GeneratingFunction and ZTransform can be expressed in terms of each other:
Linearity:
Shifting:
Convolution:
Derivative:
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