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.

Contents
1 Definitions

1.1 Ordinary generating function
1.2 Exponential generating function
1.3 Lambert series
1.4 Bell series
1.5 Dirichlet series generating functions
1.6 Polynomial sequence generating functions

2 Examples

2.1 Ordinary generating function
2.2 Exponential generating function
2.3 Bell series
2.4 Dirichlet series generating function

3 Another example
4 Uses
5 See also
6 References
7 External links

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

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

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 am,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.

Exponential generating function

The exponential generating function of a sequence an is

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

Lambert series

The Lambert series of a sequence an 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.

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.

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

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

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

DG(an;s) = ∏fp(p − s).
p

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

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 pn(x) is a sequence of polynomials and f(t) is a function of a certain form. Sheffer sequences are generated in a similar way.

Examples

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

Ordinary generating function

G(n^2;x)=\sum_{n=0}^{\infty}n^2x^n=\frac{x(x+1)}{(1-x)^3}

Exponential generating function

EG(n^2;x)=\sum _{n=0}^{\infty} \frac{n^2x^n}{n!}=x(x+1)e^x

Bell series

f_p(x)=\sum_{n=0}^\infty p^{2n}x^n=\frac{1}{1-p^2x}

Dirichlet series generating function

DG(n^2;s)=\sum_{n=1}^{\infty} \frac{n^2}{n^s}=\zeta(s-2)

Another example

Generating functions can be created by extending simpler generating functions. For example, starting with

G(1;x)=\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

and replacing x with 2x, we obtain

G(1;2x)=\frac{1}{1-2x} = 1+(2x)+(2x)^2+\ldots+(2x)^n+\ldots=G(2^n;x).

Uses

Generating functions are used to

See also

References

External links

See also: Generating function, Arithmetic function, Bell series, Binomial type, Combinatorics, Dirichlet L-series, Dirichlet character, Dirichlet series, Discrete random variable, Enumeration