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Generating function

Encyclopedia : G : GE : GEN : 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, 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.

Contents

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_^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_;x,y)=\sum_^a_x^my^n.]

Exponential generating function

The exponential generating function of a sequence an is

[EG(a_n;x)=\sum _^ a_n \frac.]

Poisson generating function

The Poisson generating function of a sequence an is

[PG(a_n;x)=\sum _^ a_n e^ \frac.]

Lambert series

The Lambert series of a sequence an is

[LG(a_n;x)=\sum _^ a_n \frac.]
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_^\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 _^ \frac.]
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(a_n;s)=\prod_ f_p(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^=\sum_^\infty 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. See the main article generalized Appell polynomials for more information.

Examples

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

Ordinary generating function

[G(n^2;x)=\sum_^n^2x^n=\frac]

Exponential generating function

[EG(n^2;x)=\sum _^ \frac=x(x+1)e^x]

Bell series

[f_p(x)=\sum_^\infty p^x^n=\frac]

Dirichlet series generating function

[DG(n^2;s)=\sum_^ \frac=\zeta(s-2)]

Another example

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

[G(1;x)=\sum_^ x^n = \frac]
and replacing [x] with [2x], we obtain

[G(1;2x)=\frac = 1+(2x)+(2x)^2+\cdots+(2x)^n+\cdots=G(2^n;x).]

More detailed example —

Consider the problem of finding a closed formula for the Fibonacci numbers Fn defined by F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2. We form the ordinary generating function

[f = \sum_ F_n X^n]
for this sequence. The generating function for the sequence (Fn−1) is Xf and that of (Fn−2) is X2f. From the recurrence relation, we therefore see that the power series Xf + X2f agrees with f except for the first two coefficients. Taking these into account, we find that

[f = Xf + X^2 f + X]
(this is the crucial step; recurrence relations can almost always be translated into equations for the generating functions). Solving this equation for f, we get

[f = \frac ]
The denominator can be factored using the golden ratio φ1 = (1 + √5)/2 and φ2 = (1 − √5)/2, and the technique of partial fraction decomposition yields

[f = \frac} - \frac} ]
These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula

[F_n = \frac } (\phi_1^n - \phi_2^n).]

Applications

Generating functions are used to

Other generating functions

Examples of polynomial sequences generated by more complex generating functions include:

See also

References

External links

 

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