gives the generating function in x for the sequence whose n series coefficient is given by the expression expr.
gives the multidimensional generating function in x1, x2, … whose n1, n2, … coefficient is given by expr.
Details and Options
- The generating function for a sequence whose n term is an 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
- In TraditionalForm, GeneratingFunction is output using .
Examplesopen allclose all
Basic Examples (3)
Basic Uses (7)
Generating function of a univariate function:
Generating function of a multivariate function:
Compute a typical generating function:
Plot the magnitude using Plot3D, ContourPlot or DensityPlot:
Generate conditions for the region of convergence:
Evaluate the generating function at a point:
Plot both the spectrum and the plot phase using color:
Plot the spectrum in the complex plane using ParametricPlot3D:
GeneratingFunction will use several properties including linearity:
Multiplication by exponentials:
Multiplication by polynomials:
GeneratingFunction automatically threads over lists:
Special Sequences (12)
Polynomials result in rational generating functions:
Factorial exponential polynomials:
Trigonometric, exponential and polynomial:
Combinations of the previous input will also generate rational generating functions:
Different ways of expressing piecewise-defined signals:
Rational exponential functions:
Hypergeometric term sequences:
The DiscreteRatio is rational for all hypergeometric term sequences:
Many functions give hypergeometric terms:
Any products are hypergeometric terms:
Generating functions of hypergeometric terms:
A holonomic sequence is defined by a linear difference equation:
Many special function are holonomic sequences in their index:
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:
Properties & Relations (5)
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:
GeneratingFunction is closely related to ExponentialGeneratingFunction:
Possible Issues (1)
A GeneratingFunction may not converge for all values of parameters:
Use GenerateConditions to get the region of convergence:
Wolfram Research (2008), GeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GeneratingFunction.html.
Wolfram Language. 2008. "GeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GeneratingFunction.html.
Wolfram Language. (2008). GeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeneratingFunction.html