Generating function

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This article is about generating functions in mathematics. For generating functions in classical mechanics, see Generating function (physics).

In mathematics a generating function is a formal power series whose coefficients encode information about a sequence a_n 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.

1 Definitions
2 Examples
3 Applications
4 Other generating functions
5 See also
6 References
7 External links

[edit] Definitions

A generating function is a clothesline on which we hang up a sequence of numbers for display.

— Herbert Wilf, Generatingfunctionology (1994)

[edit] Ordinary generating function

The ordinary generating function of a sequence a_n is

$G(a_n;x)=\sum_{n=0}^{\infty}a_nx^n.$

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

If a_n 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 a_m,n (where n and m are natural numbers) is

$G(a_{m,n};x,y)=\sum_{m,n=0}^{\infty}a_{m,n}x^my^n.$

[edit] Exponential generating function

The exponential generating function of a sequence a_n is

$EG(a_n;x)=\sum _{n=0}^{\infty} a_n \frac{x^n}{n!}.$

[edit] Poisson generating function

The Poisson generating function of a sequence a_n is

$PG(a_n;x)=\sum _{n=0}^{\infty} a_n e^{-x} \frac{x^n}{n!}.$

[edit] Lambert series

The Lambert series of a sequence a_n is

$LG(a_n;x)=\sum _{n=1}^{\infty} a_n \frac{x^n}{1-x^n}.$

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

[edit] Bell series

The Bell series of an arithmetic function f(n) and a prime p is

$f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.$

[edit] 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 a_n is

$DG(a_n;s)=\sum _{n=1}^{\infty} \frac{a_n}{n^s}.$

The Dirichlet series generating function is especially useful when a_n is a multiplicative function, when it has an Euler product expression in terms of the function's Bell series

$DG(a_n;s)=\prod_{p} f_p(p^{-s})\,.$

If a_n is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.

[edit] Polynomial sequence generating functions

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by

$e^{xf(t)}=\sum_{n=0}^\infty {p_n(x) \over n!}t^n$

where p_n(x) is a sequence of polynomials and f(t) is a function of a certain form. Sheffer sequences are generated in a similar way. See the main article generalized Appell polynomials for more information.