generating

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, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; 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 the term 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 generalized to sequences with multiple indexes. For example, the ordinary generating function of a sequence am,n (where n and m are natural numbers) is

 
Exponential generating function
The exponential generating function of a sequence an is

 Poisson generating function
The Poisson generating function of a sequence an is


 

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