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GeneratingFunction

GeneratingFunction
gives the generating function in x for the sequence whose n^(th) series coefficient is given by the expression expr.
GeneratingFunction
gives the multidimensional generating function in , , ... whose , , ... coefficient is given by expr.
  • The generating function for a sequence whose n^(th) term is is given by .
  • 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
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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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:
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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