# ExponentialGeneratingFunction

ExponentialGeneratingFunction[expr,n,x]

gives the exponential generating function in x for the sequence whose n 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  term is is given by .
• The multidimensional exponential 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

# Examples

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## Basic Examples(1)

The exponential generating function for the sequence whose n term is 1:

The  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:

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:

## 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)