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ExponentialGeneratingFunction

ExponentialGeneratingFunction[expr, n, x]
gives the exponential generating function in x for the sequence whose n^(th) term is given by the expression expr.
ExponentialGeneratingFunction[expr, {n1, n2, ...}, {x1, x2, ...}]
gives the multidimensional exponential generating function in x1, x2, ... whose n1, n2, ... term is given by expr.
  • The exponential generating function for a sequence whose n^(th) term is an is given by sum_(n=0)^(infty)a_n x^n/n!.
  • 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
EXAMPLESCLOSE ALL
Basic Examples   (1)
The exponential generating function for the sequence whose n^(th) term is 1:
The n^(th) term in the series is 1/n!:
The exponential generating function for the sequence whose n^(th) term is 1:
In[1]:=
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Out[1]=
The n^(th) term in the series is 1/n!:
In[2]:=
Click for copyable input
Out[2]=
Scope   (7)
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:
Generalizations & Extensions   (1)
Compute the exponential generating function at a point:
Options   (6)
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 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 non zero:
In addition setting GenerateConditions to True will display the conditions for convergence:
Properties & Relations   (2)
ExponentialGeneratingFunction effectively computes an infinite sum:
Linearity:
Derivative:
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