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:

Plot the complex phase:

Generate conditions for the region of convergence:

Plot the region for :

Evaluate the generating function at a point:

Plot the spectrum:

The phase:

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:

Shifts:

Multiplication by exponentials:

Multiplication by polynomials:

Conjugate:

GeneratingFunction automatically threads over lists:

Equations:

Rules:

TraditionalForm typesetting:

Discrete impulses:

Discrete unit steps:

Discrete ramps:

Polynomials result in rational generating functions:

Factorial polynomials:

Exponential functions:

Exponential polynomials:

Factorial exponential polynomials:

Trigonometric functions:

Trigonometric, exponential and polynomial:

Combinations of the previous input will also generate rational generating functions:

Different ways of expressing piecewise-defined signals:

Rational functions:

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:

Holonomic sequences:

A holonomic sequence is defined by a linear difference equation:

Many special function are holonomic sequences in their index:

Special sequences:

Periodic sequences:

Multivariate generating functions:

Linearity:

Differences and shifts:

Sums:

Integrals:

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:

Different methods may produce different results:

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: