Generating function

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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.

Contents 1 Definitions 1.1 Ordinary generating function 1.2 Exponential generating function 1.3 Poisson generating function 1.4 Lambert series 1.5 Bell series 1.6 Dirichlet series generating functions 1.7 Polynomial sequence generating functions 2 Examples 2.1 Ordinary generating function 2.2 Exponential generating function 2.3 Bell series 2.4 Dirichlet series generating function 3 Another example 4 More detailed example — Fibonacci numbers 5 Applications 6 Other generating functions 7 See also 8 References 9 External links

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

<math>G(a_n;x)=\sum_{n=0}^{\infty}a_nx^n.</math>

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

<math>G(a_{m,n};x,y)=\sum_{m,n=0}^{\infty}a_{m,n}x^my^n.</math>

Exponential generating function

The exponential generating function of a sequence a_n is

<math>EG(a_n;x)=\sum _{n=0}^{\infty} a_n \frac{x^n}{n!}.</math>

Poisson generating function

The Poisson generating function of a sequence a_n is

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

Lambert series

The Lambert series of a sequence a_n is

<math>LG(a_n;x)=\sum _{n=1}^{\infty} a_n \frac{x^n}{1-x^n}.</math>

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

<math>f_p(x)=\sum_{n=0}^\infty f(p^n)x^n.</math>

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

<math>DG(a_n;s)=\sum _{n=1}^{\infty} \frac{a_n}{n^s}.</math>

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

<math>DG(a_n;s)=\prod_{p} f_p(p^{-s})\,.</math>

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

<math>e^{xf(t)}=\sum_{n=0}^\infty {p_n(x) \over n!}t^n</math>

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

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

Ordinary generating function

<math>G(n^2;x)=\sum_{n=0}^{\infty}n^2x^n=\frac{x(x+1)}{(1-x)^3}</math>

Exponential generating function

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

Bell series

<math>f_p(x)=\sum_{n=0}^\infty p^{2n}x^n=\frac{1}{1-p^2x}</math>

Dirichlet series generating function

<math>DG(n^2;s)=\sum_{n=1}^{\infty} \frac{n^2}{n^s}=\zeta(s-2)</math>

Another example

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

<math>G(1;x)=\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}</math>

and replacing <math>x</math> with <math>2x</math>, we obtain

<math>G(1;2x)=\frac{1}{1-2x} = 1+(2x)+(2x)^2+\cdots+(2x)^n+\cdots=G(2^n;x).</math>

More detailed example — Fibonacci numbers

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

<math>

f = \sum_{n \ge 0} F_n X^n </math>

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

<math>

f = Xf + X^2 f + X </math>

(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

<math>

f = \frac{X} {1 - X - X^2} </math>

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

<math>

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

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

<math>

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

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.

Other generating functions

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

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

See also

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

References

• 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.

External links

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

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