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Encyclopedia > 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...


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. In mathematics, closed form can mean: a finitary expression, rather than one involving (for example) an infinite series, or use of recursion - this meaning usually occurs in a phrase like solution in closed form and one also says closed formula; a closed differential form: see Closed and exact differential forms. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

Contents

Definitions

A generating function is a clothesline on which we hang up a sequence of numbers for display.
Herbert Wilf, Generatingfunctionology (1994)

Herbert Wilf (born 1931) is a mathematician, specializing in combinatorics. ...

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. In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ... In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... In probability theory, the probability-generating function of a discrete random variable is a penis-like representation (the generating function) of the probability mass function of the random variable of nipples. ...


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!}.

Poisson generating function

The Poisson generating function of a sequence an is

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

Lambert series

The Lambert series of a sequence an is A Lambert series, named after Johann Heinrich Lambert, is a series taking the form It can be resummed by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function Since this last sum is a typical number-theortic sum...

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 In mathematics, the Bell series is a formal power series used to study properties of multiplicative arithmetical functions. ... In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ...

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 In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...

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 In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series. ...

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

If an is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series. In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ... In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...


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 Definition In mathematics, a polynomial sequence, i. ...

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. See the main article generalized Appell polynomials for more information. In mathematics, a polynomial sequence, i. ... In mathematics, a polynomial sequence has a generalized Appell representation if the generating function for the polynomials takes on a certain form: where the generating function or kernel is composed of the series with and and all and with Given the above, it is not hard to show that is...


Examples

Generating functions for the sequence of square numbers an = n2 are: In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ...


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+cdots+(2x)^n+cdots=G(2^n;x).

A more detailed example — Fibonacci numbers

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 In mathematics, the Fibonacci numbers form a sequence defined recursively by: That is, after two starting values, each number is the sum of the two preceding num,bers. ...

f = sum_{n ge 0} 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{x} {1 - x - x^2} .

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 The golden section is a line segment sectioned into two according to the golden ratio. ... Partial fraction decompostion is a theorem in algebra which says that a rational function can be decomposed into a polynomial plus a sum of proper fractions, each of which is either a constant over a power of a linear polynomial or a linear polynomial over a power of an irreducible...

f = frac{1}{sqrt{5}} left (frac{1}{1-phi_1 x} - frac{1} {1- phi_2 x} right ) .

These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

F_n = frac{1} {sqrt{5}} (phi_1^n - phi_2^n).

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.

In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... suvodip ... Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...

Other generating functions

Examples of polynomial sequences generated by more complex generating functions include: In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...

In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selbergs polynomials, and the Stirling interpolation polynomials as special cases. ... In mathematics, a polynomial sequence has a generalized Appell representation if the generating function for the polynomials takes on a certain form: where the generating function or kernel is composed of the series with and and all and with Given the above, it is not hard to show that is... In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. ...

See also

In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the probability-generating function of a discrete random variable is a penis-like representation (the generating function) of the probability mass function of the random variable of nipples. ... In combinatorial mathematics, Stanleys reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone and the generating function of the cones interior. ...

References

  • 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, Oren Parashnik, Concrete Mathematics. A foundation for computer science (Second Edition) Addison-Wesley. ISBN 0-201-55802-5. Chapter 7: Generating Functions, pp. 320–380

External links


  Results from FactBites:
 
Generating Functions from Interactive Mathematics Miscellany and Puzzles (757 words)
A generating function is a clothesline on which we hang up a sequence of numbers for display.
In this sense, the term generating function is not quite consistent in that the operation of substituting a number for x in order to establish the value of the series at that point, is not legal.
is the generating function of the sequence of the binomial coefficients
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The mathematical method of generating functions is used to show that the likelihood of long matches can be substantially reduced by using the tiebreak game in the fifth set, or more effectively by using a new type of game, the 50-40 game, throughout the match.
The cumulant generating function (taking the natural logarithm of the moment generating function), can also be used to calculate the parameters of the distribution in a tennis match.
The cumulant generating function is particularly useful for calculating the parameters of distributions for the number of points in a tiebreaker match, since the critical property of cumulant generating functions is that they are additive for linear combinations of independent random variables.
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