 Generating function

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 in one indeterminate, whose coefficients encode information about a sequence of numbers a_{n} that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.^{[1]} One can generalize to formal power series in more than one indeterminate, to encode information about arrays of numbers indexed by several 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 in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, 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 (rather than as a series), by some expression involving operations defined for formal power series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal power series as its Taylor series; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal power series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal power series; negative and fractional powers of x are examples of this.
Generating functions are not functions in the formal sense of a mapping from a domain to a codomain; the name is merely traditional, and they are sometimes more correctly called generating series.^{[2]}
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 a_{n} is
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 probabilitygenerating function.
The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a twodimensional array a_{m, n} (where n and m are natural numbers) is
Exponential generating function
The exponential generating function of a sequence a_{n} is
Poisson generating function
The Poisson generating function of a sequence a_{n} is
Lambert series
The Lambert series of a sequence a_{n} is
Note that in a Lambert series the index n starts at 1, not at 0.
Bell series
The Bell series of a sequence a_{n} is an expression in terms of both an indeterminate x and a prime p and is given by
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
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
If a_{n} is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet Lseries.
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
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.
Ordinary generating functions
Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the Poincaré polynomial, and others.
A key generating function is the constant sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., whose ordinary generating function is
The lefthand side is the Maclaurin series expansion of the righthand side. Alternatively, the righthand side expression can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1, in other words that all coefficients vanish, except the one of x^{0}. Moreover there can be no other power series with this property. The lefthand side therefore designates the multiplicative inverse of 1 − x in the ring of power series.
Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution x → ax gives the generating function for the geometric sequence 1,a,a^{2},a^{3},... for any constant a:
(The equality also follows directly from the fact that the lefthand side is the Maclaurin series expansion of the righthand side.) In particular,
One 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
By squaring the initial generating function, or by finding the derivative of both sides with respect to x, one sees that the coefficients form the sequence 1, 2, 3, 4, 5, ..., so one has
and the third power has as coefficients the triangular numbers 1, 3, 6, 10, 15, 21, ... whose term n is the binomial coefficient , so that
More generally, for any positive integer k, it is true that
Note that, since
one can find the ordinary generating function for the sequence 0, 1, 4, 9, 16, ... of square numbers by linear combination of binomialcoefficient generating sequences;
Rational functions
Main article: Linear recursive sequenceThe ordinary generating function of a sequence can be expressed as a rational function (the ratio of two polynomials) if and only if the sequence is a linear recursive sequence; this generalizes the examples above.
Multiplication yields convolution
Main article: Cauchy productMultiplication of ordinary generating functions yields a discrete convolution (the Cauchy product) of the sequences. For example, the sequence of cumulative sums of a sequence with ordinary generating function G(a_{n}; x) has the generating function because 1/(1x) is the ordinary generating function for the sequence (1, 1, ...).
Relation to discretetime Fourier transform
Main article: Discretetime Fourier transformWhen the series converges absolutely, is the discretetime Fourier transform of the sequence a_{0}, a_{1}, ....
Asymptotic growth of a sequence
In calculus, often the growth rate of the coefficients of a power series can be used to deduce a radius of convergence for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the asymptotic growth of the underlying sequence.
For instance, if an ordinary generating function G(a_{n}; x) that has a finite radius of convergence of r can be written as
where A(x) and B(x) are functions that are analytic to a radius of convergence greater than r (or are entire), and where B(r) ≠ 0 then
using the Gamma function.
Asymptotic growth of the sequence of squares
As derived above, the ordinary generating function for the sequence of squares is With r = 1, α = 0, β = 3, A(x) = 0, and B(x) = x(x+1), we can verify that the squares grow as expected, like the squares:
Asymptotic growth of the Catalan numbers
Main article: Catalan numberThe ordinary generating function for the Catalan numbers is With r = 1/4, α = 1, β = −1/2, A(x) = 1/2, and B(x) = −1/2, we can conclude that, for the Catalan numbers,
Bivariate and multivariate generating functions
One can define generating functions in several variables for arrays with several indices. These are called multivariate generating functions or, sometimes, super generating functions. For two variables, these are often called bivariate generating functions.
For instance, since (1 + x)^{n} is the ordinary generating function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients for all k and n. To do this, consider (1 + x)^{n} as itself a series, in n, and find the generating function in y that has these as coefficients. Since the generating function for a^{n} is 1 / (1 − ay), the generating function for the binomial coefficients is:
Examples
Main article: Examples of generating functionsGenerating functions for the sequence of square numbers a_{n} = n^{2} are:
Ordinary generating function
Exponential generating function
Bell series
Dirichlet series generating function
using the Riemann zeta function.
The sequence a_{n} generated by a Dirichlet series generating function corresponding to:
where ζ(s) is the Riemann zeta function, has the ordinary generating function:
Multivariate generating function
Multivariate generating functions arise in practice when calculating the number of contingency tables of nonnegative integers with specified row and column totals. Suppose the table has r rows and c columns; the row sums are and the column sums are . Then, according to I. J. Good,^{[3]} the number of such tables is the coefficient of in
Applications
Generating functions are used to
 Find a closed formula for a sequence given in a recurrence relation. For example consider Fibonacci numbers.
 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 may be related.
 Explore the asymptotic behaviour of sequences.
 Prove identities involving sequences.
 Solve enumeration problems in combinatorics and encoding their solutions. Rook polynomials are an example of an application in combinatorics.
 Evaluate infinite sums.
Other generating functions
Examples of polynomial sequences generated by more complex generating functions include:
 Appell polynomials
 Chebyshev polynomials
 Difference polynomials
 Generalized Appell polynomials
 Qdifference polynomials
Similar concepts
Polynomial interpolation is finding a polynomial whose values (not coefficients) agree with a given sequence; the Hilbert polynomial is an abstract case of this in commutative algebra.
See also
 Momentgenerating function
 Probabilitygenerating function
 Stanley's reciprocity theorem
 Applications to partitions
 Combinatorial principles
References
 Doubilet, Peter; Rota, GianCarlo; Stanley, Richard (1972). "On the foundations of combinatorial theory. VI. The idea of generating function". Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 2: 267–318. http://projecteuclid.org/euclid.bsmsp/1200514223.
 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (1994). Concrete Mathematics. A foundation for computer science (second ed.). AddisonWesley. pp. 320–380. ISBN 0201558025.
 Herbert S. Wilf (1994). Generatingfunctionology (second ed.). Academic Press. ISBN 0127519564. http://www.math.upenn.edu/%7Ewilf/DownldGF.html.
 Flajolet, Philippe; Sedgewick, Robert (2009). Analytic Combinatorics. Cambridge University Press. ISBN 9780521898065. http://algo.inria.fr/flajolet/Publications/book.pdf.
 ^ Donald E. Knuth, The Art of Computer Programming, Volume 1 Fundamental Algorithms (Third Edition) AddisonWesley. ISBN 0201896834. Section 1.2.9: Generating Functions, pp. 86
 ^ This alternative term can already be found in E.N. Gilbert, Enumeration of Labeled graphs, Canadian Journal of Mathematics 3, 1956, p. 405–411, but its use is rare before the year 2000; since then it appears to be increasing
 ^ Good, I. J. (1986). "On applications of symmetric Dirichlet distributions and their mixtures to contingency tables". The Annals of Statistics 4 (6): 1159–1189. doi:10.1214/aos/1176343649.
External links
 Generating Functions, Power Indices and Coin Change at cuttheknot
 Generatingfunctionology PDF download page
 (French) 1031 Generating Functions
 Ignacio Larrosa Cañestro, LeónSotelo, Marko Riedel, Georges Zeller, Suma de números equilibrados, newsgroup es.ciencia.matematicas
 Frederick Lecue; Riedel, Marko, et al., Permutation, LesMathematiques.net, in French, title somewhat misleading.
 "Generating Functions" by Ed Pegg, Jr., Wolfram Demonstrations Project, 2007.
Categories: Generating functions
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