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GeneratingFunction

GeneratingFunction[expr, n, x]
gives the generating function in x for the sequence whose n^(th) series coefficient is given by the expression expr.
GeneratingFunction[expr, {n1, n2, ...}, {x1, x2, ...}]
gives the multidimensional generating function in x1, x2, ... whose n1, n2, ... coefficient is given by expr.
  • The generating function for a sequence whose n^(th) term is an is given by sum_(n=0)^(infty)a_n x^n.
  • The multidimensional generating function is given by .
  • The following options can be given:
Assumptions$Assumptionsassumptions to make about parameters
GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
MethodAutomaticmethod to use
VerifyConvergenceTruewhether to verify convergence
EXAMPLESCLOSE ALL
Basic Examples   (3)
The generating function for the sequence whose n^(th) 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^(th) term is 1:
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All coefficients in the series are 1:
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Univariate generating function:
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Multivariate:
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The generating function for a shifted sequence:
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Scope   (4)
Polynomials can be expressed in terms of rational functions:
Rational generating functions:
Generating function of a periodic sequence:
Special functions:
Generalizations & Extensions   (1)
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 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 non zero:
In addition setting GenerateConditions to True will display the conditions for convergence:
Properties & Relations   (4)
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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