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.

Details and Options

  • 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

Examples

open allclose all

Basic Examples  (1)

The exponential generating function for the sequence whose n^(th) term is 1:

The ^(th) term in the series is :

Scope  (19)

Basic Uses  (6)

Exponential generating function of a univariate sequence:

Exponential generating function of a multivariate sequence:

Compute a typical exponential 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 exponential 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:

ExponentialGeneratingFunction will use several properties including linearity:

Multiplication by exponentials:

Multiplication by polynomials:

Conjugate:

ExponentialGeneratingFunction automatically threads over lists:

Equations:

Rules:

Special Sequences  (13)

Discrete unit steps:

Discrete ramps:

Polynomials:

Factorial polynomials:

Exponential functions:

Exponential polynomials:

Factorial exponential polynomials:

Trigonometric functions:

Trigonometric, exponential and polynomial:

Combinations of the previous input:

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:

Transforms of hypergeometric terms:

Holonomic sequences:

A holonomic sequence is defined by a linear difference equation:

Many special function are holonomic sequences in their index:

DifferenceRoot in general results in DifferentialRoot functions:

Special sequences:

Periodic sequences:

Multivariate exponential generating functions:

Generalizations & Extensions  (1)

Compute the exponential generating function at a point:

Options  (5)

GenerateConditions  (1)

By default, no conditions are given for where a generating function is convergent:

Use GenerateConditions to generate conditions of validity:

Method  (1)

Different methods may produce different formulas:

VerifyConvergence  (3)

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  (3)

ExponentialGeneratingFunction effectively computes an infinite sum:

Linearity:

ExponentialGeneratingFunction is closely related to GeneratingFunction:

ZTransform:

FourierSequenceTransform:

Possible Issues  (1)

A ExponentialGeneratingFunction may not converge for all values of parameters:

Use GenerateConditions to get the region of convergence:

Neat Examples  (1)

Create a gallery of exponential generating functions:

Introduced in 2008
 (7.0)