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| {{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|signalling molecule|Epidermal growth factor|Generator in computer programming|Generator (computer programming)}}
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| In [[mathematics]], a '''generating function''' is a [[formal power series]] in one [[Indeterminate (variable)|indeterminate]], whose [[coefficient]]s encode information about a [[sequence]] of numbers ''a''<sub>''n''</sub> that is [[Indexed family|indexed]] by the [[natural number]]s. Generating functions were first introduced by [[Abraham de Moivre]] in 1730, in order to solve the general linear recurrence problem.<ref>[[Donald Knuth|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. 86</ref> One can generalize to formal power series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural numbers. | |
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| 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.
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| Generating functions are often expressed in [[Closed-form expression|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.
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| Generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]; the name is merely traditional, and they are sometimes more correctly called '''generating series'''.<ref>This alternative term can already be found in E.N. Gilbert, ''Enumeration of Labeled graphs'', Canadian Journal of Mathematics 3, 1956, [http://books.google.fr/books?id=x34z99fCRbsC&lpg=PA405&ots=eOp9p9mIoD&dq=%22generating%20series%22&lr=lang_en&pg=PA407#v=onepage&q=%22generating%20series%22&f=false p. 405–411], but its use is rare before the year 2000; since then it appears to be increasing</ref>
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| ==Definitions==
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| :''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
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| :—[[Herbert Wilf]], ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)
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| ===Ordinary generating function===
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| The ''ordinary generating function'' of a sequence ''a''<sub>''n''</sub> is
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| :<math>G(a_n;x)=\sum_{n=0}^\infty a_nx^n.</math>
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| When the term ''generating function'' is used without qualification, it is usually taken to mean an ordinary generating function.
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| If ''a''<sub>''n''</sub> is the [[probability mass function]] of a [[discrete random variable]], then its ordinary generating function is called a [[probability-generating function]].
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| The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array ''a''<sub>''m, n''</sub> (where ''n'' and ''m'' are natural numbers) is
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| :<math>G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n}x^my^n.</math>
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| ===Exponential generating function===
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| The ''exponential generating function'' of a sequence ''a''<sub>''n''</sub> is
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| :<math>\operatorname{EG}(a_n;x)=\sum _{n=0}^{\infty} a_n \frac{x^n}{n!}.</math>
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| Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{cite book
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| | last1 = Flajolet | first1 = Philippe | author1-link = Philippe Flajolet
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| | last2 = Sedgewick | first2 = Robert | author2-link = Robert Sedgewick (computer scientist)
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| | page = 95
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| | publisher = Cambridge University Press
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| | title = Analytic Combinatorics
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| | url = http://algo.inria.fr/flajolet/Publications/book.pdf
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| | year = 2009}}</ref>
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| ===Poisson generating function===
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| The ''Poisson generating function'' of a sequence ''a''<sub>''n''</sub> is
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| :<math>\operatorname{PG}(a_n;x)=\sum _{n=0}^{\infty} a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x)\,.</math>
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| ===Lambert series===
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| The ''[[Lambert series]]'' of a sequence ''a''<sub>''n''</sub> is
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| :<math>\operatorname{LG}(a_n;x)=\sum _{n=1}^{\infty} a_n \frac{x^n}{1-x^n}.</math>
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| Note that in a Lambert series the index ''n'' starts at 1, not at 0, as the first term would otherwise be undefined.
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| ===Bell series===
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| The [[Bell series]] of a sequence ''a''<sub>''n''</sub> is an expression in terms of both an indeterminate ''x'' and a prime ''p'' and is given by<ref name=A4243>Apostol (1976) pp.42–43</ref> | |
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| :<math>\operatorname{BG}_p(a_n;x)=\sum_{n=0}^\infty a_{p^n}x^n.</math>
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| ===Dirichlet series generating functions===
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| [[Formal 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''<sub>''n''</sub> is<ref name=W56>Wilf (1994) p.56</ref>
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| :<math>\operatorname{DG}(a_n;s)=\sum _{n=1}^{\infty} \frac{a_n}{n^s}.</math>
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| The Dirichlet series generating function is especially useful when ''a''<sub>''n''</sub> is a [[multiplicative function]], when it has an [[Euler product]] expression<ref name=W59>Wilf (1994) p.59</ref> in terms of the function's Bell series
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| :<math>\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.</math> | |
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| If ''a''<sub>''n''</sub> is a [[Dirichlet character]] then its Dirichlet series generating function is called a [[Dirichlet L-series]].
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| ===Polynomial sequence generating functions===
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| The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of [[binomial type]] are generated by
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| :<math>e^{xf(t)}=\sum_{n=0}^\infty {p_n(x) \over n!}t^n</math>
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| where ''p''<sub>''n''</sub>(''x'') is a sequence of polynomials and ''f''(''t'') is a function of a certain form. [[Sheffer sequence]]s are generated in a similar way. See the main article [[generalized Appell polynomials]] for more information.
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| == Ordinary generating functions ==
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| 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.
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| A key generating function is the constant sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., whose ordinary generating function is
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| :<math>\sum_{n=0}^{\infty}x^n={1\over1-x}.</math>
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| The left-hand side is the [[Maclaurin series]] expansion of the right-hand side. Alternatively, the right-hand 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 except the one of ''x''<sup>0</sup> vanish. Moreover there can be no other power series with this property. The left-hand side therefore designates the [[multiplicative inverse]] of 1 − ''x'' in the ring of power series.
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| 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 progression|geometric sequence]] 1,''a'',''a''<sup>2</sup>,''a''<sup>3</sup>,... for any constant ''a'':
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| :<math>\sum_{n=0}^{\infty}(ax)^n={1\over1-ax}\,.</math>
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| (The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,
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| :<math>\sum_{n=0}^{\infty}(-1)^nx^n={1\over1+x}\,.</math>
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| 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
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| :<math>\sum_{n=0}^{\infty}x^{2n}={1\over1-x^2}.</math>
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| By squaring the initial generating function, or by finding the derivative of both sides with respect to ''x'' and making a change of running variable ''n'' → ''n-1'', one sees that the coefficients form the sequence 1, 2, 3, 4, 5, ..., so one has
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| :<math>\sum_{n=0}^{\infty}(n+1)x^n={1\over(1-x)^2},</math>
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| and the third power has as coefficients the [[triangular number]]s 1, 3, 6, 10, 15, 21, ... whose term ''n'' is the [[binomial coefficient]] <math>\tbinom{n+2}2</math>, so that
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| :<math>\sum_{n=0}^{\infty}\tbinom{n+2}2 x^n={1\over(1-x)^3}.</math>
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| More generally, for any positive integer ''k'', it is true that
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| :<math>\sum_{n=0}^{\infty}\tbinom{n+k}k x^n={1\over(1-x)^{k+1}}.</math>
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| Note that, since
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| :<math>2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0= 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2\,,</math>
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| one can find the ordinary generating function for the sequence 0, 1, 4, 9, 16, ... of [[square number]]s by linear combination of binomial-coefficient generating sequences;
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| :<math>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}.</math>
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| === Rational functions ===
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| {{Main|Linear recursive sequence}}
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| The 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]] with constant coefficients; this generalizes the examples above. Going in the reverse direction, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off).
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| === Multiplication yields convolution ===
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| {{Main|Cauchy product}}
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| Multiplication of ordinary generating functions yields a discrete [[convolution]] (the [[Cauchy product]]) of the sequences. For example, the sequence of cumulative sums <math>(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math> of a sequence with ordinary generating function G(''a''<sub>''n''</sub>; ''x'') has the generating function <math>G(a_n; x) \frac{1}{1-x}</math> because 1/(1-''x'') is the ordinary generating function for the sequence (1, 1, ...).
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| === Relation to discrete-time Fourier transform ===
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| {{Main|Discrete-time Fourier transform}}
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| When the series [[Absolute convergence|converges absolutely]], <math>G\left(a_n; e^{-i \omega}\right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math> is the discrete-time Fourier transform of the sequence a<sub>0</sub>, a<sub>1</sub>, ....
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| === Asymptotic growth of a sequence ===
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| 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 analysis|asymptotic growth]] of the underlying sequence.
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| For instance, if an ordinary generating function ''G''(''a''<sub>''n''</sub>; ''x'') that has a finite radius of convergence of ''r'' can be written as
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| :<math>G(a_n; x) = \frac{A(x) + B(x) (1- x/r)^{-\beta}}{x^{\alpha}} \,</math>
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| where ''A''(''x'') and ''B''(''x'') are functions that are [[analytic function|analytic]] to a radius of convergence greater than ''r'' (or are [[Entire function|entire]]), and where ''B''(''r'') ≠ 0 then
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| :<math>a_n \sim \frac{B(r)}{r^{\alpha} \Gamma(\beta)} \, n^{\beta-1}(1/r)^{n} \sim \frac{B(r)}{r^{\alpha}} \, \binom{n+\beta-1}{n \quad \beta-1}(1/r)^{n} \,,</math>
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| using the [[Gamma function]] or a [[binomial coefficient]]. Instead, if ''G'' is an exponential generating function then it is ''a''<sub>''n''</sub>/''n''! that grows according to these asymptotic formulae.
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| ==== Asymptotic growth of the sequence of squares ====
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| As derived above, the ordinary generating function for the sequence of squares is <math>\frac{x(x+1)}{(1-x)^3}\,.</math> 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:
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| :<math>a_n \sim \frac{B(r)}{r^{\alpha} \Gamma(\beta)} \, n^{\beta-1}(1/r)^{n} = \frac{1(1+1)}{1^0\,\Gamma(3)}\,n^{3-1} (1/1)^n = n^2\,.</math>
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| ==== Asymptotic growth of the Catalan numbers ====
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| {{Main|Catalan number}}
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| The ordinary generating function for the Catalan numbers is
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| <math>\frac{1-\sqrt{1-4x}}{2x}\,.</math>
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| 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,
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| :<math>a_n \sim \frac{B(r)}{r^{\alpha} \Gamma(\beta)} \, n^{\beta-1}(1/r)^{n}
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| = \frac{-1/2}{(1/4)^1 \Gamma(-1/2)} \, n^{-1/2-1} \left(\frac{1}{1/4}\right)^n
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| = \frac{n^{-3/2} \, 4^n}{\sqrt{\pi}} \,.</math>
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| === Bivariate and multivariate generating functions ===
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| 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'''.
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| For instance, since <math>(1+x)^n</math> 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 <math>\binom{n}{k}</math> for all ''k'' and ''n''.
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| To do this, consider <math>(1+x)^n</math> as itself a series, in ''n'', and find the generating function in ''y'' that has these as coefficients. Since the generating function for <math>a^n</math> is <math>1/(1-ay)</math>, the generating function for the binomial coefficients is:
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| :<math>\sum_{n,k} \binom n k x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}\,.</math>
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| ==Examples==
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| {{Main|Examples of generating functions}}
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| Generating functions for the sequence of [[square number]]s ''a''<sub>''n''</sub> = ''n''<sup>2</sup> are:
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| ===Ordinary generating function===
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| :<math>G(n^2;x)=\sum_{n=0}^{\infty}n^2x^n=\frac{x(x+1)}{(1-x)^3}</math>
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| ===Exponential generating function===
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| :<math>\operatorname{EG}(n^2;x)=\sum _{n=0}^{\infty} \frac{n^2x^n}{n!}=x(x+1)e^x</math>
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| ===Bell series===
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| :<math>\operatorname{BG}_p(n^2;x)=\sum_{n=0}^\infty (p^{n})^2x^n=\frac{1}{1-p^2x}</math>
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| ===Dirichlet series generating function===
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| :<math>\operatorname{DG}(n^2;s)=\sum_{n=1}^{\infty} \frac{n^2}{n^s}=\zeta(s-2)\,,</math>
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| using the [[Riemann zeta function]].
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| The sequence <math>a_n</math> generated by a Dirichlet series generating function corresponding to:
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| :<math>\operatorname{DG}(a_n;s)=\zeta(s)^m</math>
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| where <math>\zeta(s)</math> is the [[Riemann zeta function]], has the ordinary generating function:
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| :<math>\begin{align}
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| \sum \limits_{n=1}^{\infty} a_nx^n
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| = x &+ {m \choose 1}\sum \limits_{a=2}^{\infty} x^{a}
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| + {m \choose 2}\sum \limits_{a=2}^{\infty} \sum \limits_{b=2}^{\infty} x^{ab} \\
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| &+ {m \choose 3}\sum \limits_{a=2}^{\infty} \sum \limits_{b=2}^{\infty} \sum \limits_{c=2}^{\infty} x^{abc} + {m \choose 4}\sum \limits_{a=2}^{\infty} \sum \limits_{b=2}^{\infty} \sum \limits_{c=2}^{\infty} \sum \limits_{d=2}^{\infty} x^{abcd} +...
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| \end{align}</math>
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| ===Multivariate generating function===
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| Multivariate generating functions arise in practice when calculating the number of [[contingency tables]] of non-negative integers with specified row and column totals. Suppose the table has ''r'' rows and ''c'' columns; the row sums are <math>t_1,\ldots t_r</math> and the column sums are <math>s_1,\ldots s_c</math>. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal| doi=10.1214/aos/1176343649| last=Good| first=I. J.| title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables| journal=The Annals of Statistics| year=1986| volume=4| issue=6|pages=1159–1189| postscript=.}}</ref> the number of such tables is the coefficient of <math>x_1^{t_1}\ldots x_r^{t_r}y_1^{s_1}\ldots y_c^{s_c}</math> in
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| :<math>
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| \prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.
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| </math> | |
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| ==Applications==
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| Generating functions are used to
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| * Find a [[closed formula]] for a sequence given in a recurrence relation. For example consider [[Fibonacci number#Power_series|Fibonacci numbers]].
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| * Find [[recurrence relation]]s for sequences—the form of a generating function may suggest a recurrence formula.
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| * Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.
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| * Explore the asymptotic behaviour of sequences.
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| * Prove identities involving sequences.
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| * Solve [[enumeration]] problems in [[combinatorics]] and encoding their solutions. [[Rook polynomial]]s are an example of an application in combinatorics.
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| * Evaluate infinite sums.
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| ==Other generating functions==
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| Examples of [[polynomial sequence]]s generated by more complex generating functions include:
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| * [[Appell polynomials]]
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| * [[Chebyshev polynomials]]
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| * [[Difference polynomials]]
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| * [[Generalized Appell polynomials]]
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| * [[Q-difference polynomial]]s
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| == Similar concepts ==
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| [[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]].
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| ==See also==
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| *[[Moment-generating function]]
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| *[[Probability-generating function]]
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| *[[Stanley's reciprocity theorem]]
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| *Applications to [[Partition (number theory)|partitions]]{{which|date=August 2013}}
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| * [[Combinatorial principles]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal |title=On the foundations of combinatorial theory. VI. The idea of generating function |last1=Doubilet |first1=Peter | author1-link= |last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | last3=Stanley | first3=Richard | author3-link=Richard P. Stanley | journal=Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability |volume=2 |pages=267–318 |year=1972 | zbl=0267.05002 | url=http://projecteuclid.org/euclid.bsmsp/1200514223 |postscript=. }} Reprinted in {{cite book | last=Rota | first=Gian-Carlo | authorlink=Gian-Carlo Rota | others=With the collaboration of P. Doubilet, C. Greene, D. Kahaner, [[Andrew Odlyzko|A. Odlyzko]] and [[Richard P. Stanley|R. Stanley]] | title=Finite Operator Calculus | chapter=3. The idea of generating function | pages=83–134 | publisher=Academic Press | year=1975 | isbn=0-12-596650-4 | zbl=0328.05007 }}
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| * {{Apostol IANT}}
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| * {{cite book |author=[[Ronald Graham|Ronald L. Graham]], [[Donald Knuth|Donald E. Knuth]], and [[Oren Patashnik]] |title=[[Concrete Mathematics|Concrete Mathematics. A foundation for computer science]] |edition=second |year=1994 |publisher=Addison-Wesley |isbn=0-201-55802-5 |chapter=Chapter 7: Generating Functions |pages=320–380| zbl=0836.00001 }}
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| * {{cite book | last=Wilf | first=Herbert S. | authorlink=Herbert Wilf | title=Generatingfunctionology | edition=2nd | location=Boston, MA | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 | url=http://www.math.upenn.edu/%7Ewilf/DownldGF.html }}
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| * {{cite book | last1 = Flajolet | first1 = Philippe | authorlink1 = Philippe Flajolet | last2 = Sedgewick | first2 = Robert | authorlink2 = Robert Sedgewick (computer scientist) | title = Analytic Combinatorics | url = http://algo.inria.fr/flajolet/Publications/book.pdf | year = 2009 | publisher = Cambridge University Press | location = | isbn = 978-0-521-89806-5 | zbl=1165.05001 }}
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| ==External links==
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| * {{springer|title=Generating function|id=p/g043900}}
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| * [http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml Generating Functions, Power Indices and Coin Change] at [[cut-the-knot]]
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| * [http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology PDF download page]
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| * {{fr icon}} [http://www.lacim.uqam.ca/~plouffe/articles/FonctionsGeneratrices.pdf 1031 Generating Functions]
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| * Ignacio Larrosa Cañestro, León-Sotelo, Marko Riedel, Georges Zeller, ''[http://groups.google.com/group/es.ciencia.matematicas/browse_thread/thread/26328abc49e15dd9/88b7b522437223ce#88b7b522437223ce Suma de números equilibrados], newsgroup es.ciencia.matematicas''
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| * Frederick Lecue; Riedel, Marko, ''et al.'', [http://les-mathematiques.u-strasbg.fr/phorum5/read.php?12,360025 ''Permutation''], ''Les-Mathematiques.net'', in French, title somewhat misleading.
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| * [http://demonstrations.wolfram.com/GeneratingFunctions/ "Generating Functions"] by [[Ed Pegg, Jr.]], [[Wolfram Demonstrations Project]], 2007.
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| {{DEFAULTSORT:Generating Function}}
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| [[Category:Generating functions| ]]
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