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# Generating function

In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers.

There are various types of generating functions - definitions and examples are given below. Every sequence has a generating function of each type. The particular generating function that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in closed form as functions of a formal argument x. Sometimes a generating function is evaluated at a specific value of x. However, it must be remembered that generating functions are formal power series, and they will not necessarily converge for all values of x.

## Definitions

A generating function is a clothesline on which we hang up a sequence of numbers for display. -- Herbert Wilf, generatingfunctionology (1994)

### Ordinary generating function

The ordinary generating function of a sequence an is

When generating function is used without qualification, it is usually taken to mean an ordinary generating function.

If an is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.

The ordinary generating function can be generalised to sequences with multiple indexes. For example, the ordinary generating function of a sequence anm (where n and m are natural numbers) is

### Exponential generating function

The exponential generating function of a sequence an is

### Lambert series

The Lambert series of a sequence an is

Note that in a Lambert series the index n starts at 1, not at 0.

### Dirichlet series generating functions

Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence an is

Dirichlet series generating functions are especially useful for multiplicative functions, when they have an Euler product expression. If an is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.

## Examples

Generating functions for the sequence of square numbers an = n2 are :-

## Uses

Generating functions are used to :-