Generating function

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This article is about generating functions in mathematics. For generating functions in classical mechanics, see Generating function (physics). For Signalling molecule, see Epidermal growth factor.

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.

It is important to note that generating functions are not functions in the formal sense of a mapping from a domain to a codomain; the name merely stems from the historical study of the structures.

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 generalized 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.

[edit] Examples

Main article: Examples of generating functions

When working with generating functions, it is important to recogise the expressions of some fundamenal sequences.

[edit] Ordinary generating functions

The most fundamental of all is the constant sequence 1,1,1,1,..., whose ordinary generating function is

$\sum_{n\in\mathbf{N}}X^n={1\over1-X}.$

The right hand side expression can be justified by multiplying the power series on the left by $X - 1$ , and checking that ther result is the constant power series 1, in other words that all coefficients vanish, except the one of X⁰. (Moreover there can be no other power series with this property, since a power series ring like Z[[X]] is an integral domain.) The right hand side therefore designates the inverse of $X - 1$ in the ring of power series.

Expressions for the ordinary generating of other sequences are easily derived for this one. For instance for the geometric series 1,a,a²,a³,... for any constant a one has

$\sum_{n\in\mathbf{N}}a^nX^n={1\over1-aX},$

and in particular

$\sum_{n\in\mathbf{N}}(-1)^nX^n={1\over1+X}.$

On can also introduce regular "gaps" in the sequence by replacing X by some power, of X, so for instance for the sequence 1,0,1,0,1,0,1,0,.... one gets the generating function

$\sum_{n\in\mathbf{N}}X^{2n}={1\over1-X^2}.$

Computing the square of the initial generating function, one easily sees that the coefficients form the sequence 1,2,3,4,5,..., so one has

$\sum_{n\in\mathbf{N}}(n+1)X^n={1\over(1-X)^2},$

and the third power has as coefficients the triangular numbers 1,3,6,10,15,21,... whose term n is the binomial coefficient $\tbinom{n+2}2$ , so that

$\sum_{n\in\mathbf{N}}\tbinom{n+2}2 X^n={1\over(1-X)^3}.$

Since $\tbinom{n+2}2=\frac12{(n+1)(n+2)} =\frac12(n^2+3n+2)$ one can find the ordinary generating function for the sequence 0,1,4,9,16,... of square numbers by linear combination of the preceding sequences;

$G(n^2;x)=\sum_{n=0}^{\infty}n^2x^n={2\over(1-x)^3}-{3\over(1-x)^2}+{1\over1-x}=\frac{x(x+1)}{(1-x)^3}.$

[edit] Exponential generating function

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

[edit] Bell series

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

[edit] Dirichlet series generating function

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

[edit] Applications

Generating functions are used to

Find recurrence relations for sequences – the form of a generating function may suggest a recurrence formula.
Find relationships between sequences – if the generating functions of two sequences have a similar form, then the sequences themselves are probably related.
Explore the asymptotic behaviour of sequences.
Prove identities involving sequences.
Solve enumeration problems in combinatorics.
Evaluate infinite sums.

[edit] Other generating functions

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

[edit] See also

[edit] References

Herbert S. Wilf, Generatingfunctionology (Second Edition) (1994) Academic Press. ISBN 0-12-751956-4.

Donald E. Knuth, The Art of Computer Programming, Volume 1 Fundamental Algorithms (Third Edition) Addison-Wesley. ISBN 0-201-89683-4. Section 1.2.9: Generating Functions, pp.87–96.

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics. A foundation for computer science (Second Edition) Addison-Wesley. ISBN 0-201-55802-5. Chapter 7: Generating Functions, pp. 320–380

[edit] External links

Generating Functions, Power Indices and Coin Change at cut-the-knot
Generatingfunctionology PDF download page
1031 Generating Functions
Ignacio Larrosa Cañestro, León-Sotelo, Marko Riedel, Georges Zeller, Suma de números equilibrados, newsgroup es.ciencia.matematicas
Frederick Lecue; Riedel, Marko, et al., Permutation, Les-Mathematiques.net, in French, title somewhat misleading.
"Generating Functions" by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007.

Categories: Combinatorics | Exponentials

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Generating function

From Wikipedia, the free encyclopedia

Contents

[edit] Definitions

[edit] Ordinary generating function

[edit] Exponential generating function

[edit] Poisson generating function

[edit] Lambert series

[edit] Bell series

[edit] Dirichlet series generating functions

[edit] Polynomial sequence generating functions

[edit] Examples

[edit] Ordinary generating functions

[edit] Exponential generating function

[edit] Bell series

[edit] Dirichlet series generating function

[edit] Applications

[edit] Other generating functions

[edit] See also

[edit] References

[edit] External links

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