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.

Definitions

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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

$G(a\_n;x)=\backslash sum\_\{n=0\}^\{\backslash infty\}a\_nx^n.$

When 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 generalised 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

$G(a\_\{m,n\};x,y)=\backslash sum\_\{m,n=0\}^\{\backslash infty\}a\_\{m,n\}x^my^n.$

Exponential generating function

The exponential generating function of a sequence a_n is

$EG(a\_n;x)=\backslash sum\; \_\{n=0\}^\{\backslash infty\}\; a\_n\; \backslash frac\{x^n\}\{n!\}.$

Poisson generating function

The Poisson generating function of a sequence a_n is

$PG(a\_n;x)=\backslash sum\; \_\{n=0\}^\{\backslash infty\}\; a\_n\; e^\{-x\}\; \backslash frac\{x^n\}\{n!\}.$

Lambert series

The Lambert series of a sequence a_n is

$LG(a\_n;x)=\backslash sum\; \_\{n=1\}^\{\backslash infty\}\; a\_n\; \backslash frac\{x^n\}\{1-x^n\}.$

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

$f\_p(x)=\backslash sum\_\{n=0\}^\backslash infty\; f(p^n)x^n.$

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 a_n is

$DG(a\_n;s)=\backslash sum\; \_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a\_n\}\{n^s\}.$

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

$DG(a\_n;s)=\backslash prod\_\{p\}\; f\_p(p^\{-s\})\backslash ,.$

If a_n 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

$e^\{xf(t)\}=\backslash sum\_\{n=0\}^\backslash infty\; \{p\_n(x)\; \backslash over\; n!\}t^n$

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.

Examples

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

Ordinary generating function

$G(n^2;x)=\backslash sum\_\{n=0\}^\{\backslash infty\}n^2x^n=\backslash frac\{x(x+1)\}\{(1-x)^3\}$

Exponential generating function

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

Bell series

$f\_p(x)=\backslash sum\_\{n=0\}^\backslash infty\; p^\{2n\}x^n=\backslash frac\{1\}\{1-p^2x\}$

Dirichlet series generating function

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

Another example

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

$G(1;x)=\backslash sum\_\{n=0\}^\{\backslash infty\}\; x^n\; =\; \backslash frac\{1\}\{1-x\}$

and replacing $x$ with $2x$, we obtain

$G(1;2x)=\backslash frac\{1\}\{1-2x\}\; =\; 1+(2x)+(2x)^2+\backslash cdots+(2x)^n+\backslash cdots=G(2^n;x).$

More detailed example — Fibonacci numbers

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

$$

f = \sum_{n \ge 0} F_n X^n

for this sequence. The generating function for the sequence (F_n−1) is Xf and that of (F_n−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

$$

f = \frac{1 / \sqrt{5}} {1-\phi_1 X} - \frac{1/\sqrt{5}} {1- \phi_2 X}

These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula

$$

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

Applications

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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.

Other generating functions

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

• difference polynomials
• generalized Appell polynomials
• q-difference polynomials

See also

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• Moment-generating function
• Probability-generating function
• Stanley's reciprocity theorem

References

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• Herbert S. Wilf, Generatingfunctionology (Second Edition) (1994) Academic Press. ISBN 0127519564.

• Donald E. Knuth, The Art of Computer Programming, Volume 1 Fundamental Algorithms (Third Edition) Addison-Wesley. ISBN 020189683-4. 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 0201558025. Chapter 7: Generating Functions, pp. 320–380

External links

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• Generating Functions, Power Indices and Coin Change at cut-the-knot
• Generatingfunctionology (PDF)

Combinatorics | Exponentials

Erzeugende Funktion | Fonction génératrice | פונקציה יוצרת | Funzione generatrice | Funkcja tworząca | Производящая функция