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ExponentialGeneratingFunction

ExponentialGeneratingFunction
gives the exponential generating function in x for the sequence whose n^(th) term is given by the expression expr.
ExponentialGeneratingFunction
gives the multidimensional exponential generating function in , , ... whose , , ... term is given by expr.
  • The exponential generating function for a sequence whose ^(th) term is is given by .
  • The multidimensional exponential 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 exponential generating function for the sequence whose n^(th) term is 1:
The ^(th) term in the series is :
The exponential generating function for the sequence whose n^(th) term is 1:
In[1]:=
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Out[1]=
The ^(th) term in the series is :
In[2]:=
Click for copyable input
Out[2]=
Univariate exponential generating function:
Multivariate:
Periodic sequences:
Polynomial:
Rational function:
Polynomial exponential:
Polynomial trigonometric:
Hypergeometric terms:
Special functions:
DifferenceRoot in general results in DifferentialRoot functions:
Compute the exponential generating function at a point:
In general this generating function cannot be given:
By providing additional Assumptions, an equivalent form can be given:
By default no conditions are given for where a generating function is convergent:
Use GenerateConditions to generate conditions of validity:
Different methods may produce different formulas:
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:
ExponentialGeneratingFunction effectively computes an infinite sum:
Linearity:
Derivative:
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