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In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an 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. 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.
The ordinary generating function of a sequence an is
When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function.
The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array am, n (where n and m are natural numbers) is
The exponential generating function of a sequence an is
The Poisson generating function of a sequence an is
The Lambert series of a sequence an is
Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.
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 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.
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 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 x0 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.
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, a2, a3, ... for any constant a:
(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,
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 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
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 binomial-coefficient generating sequences;
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).
of a sequence with ordinary generating function G(an; x) has the generating function
because 1/(1-x) is the ordinary generating function for the sequence (1, 1, ...).
When the series converges absolutely,
is the discrete-time Fourier transform of the sequence a0, a1, ....
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(an; x) that has a finite radius of convergence of r can be written as
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:
The 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,
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 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 as itself a series, in n, and find the generating function in y that has these as coefficients. Since the generating function for is
the generating function for the binomial coefficients is:
Generating functions for the sequence of square numbers an = n2 are:
using the Riemann zeta function.
The sequence an generated by a Dirichlet series generating function corresponding to:
where is the Riemann zeta function, has the ordinary generating function:
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 and the column sums are . Then, according to I. J. Good, the number of such tables is the coefficient of
Generating functions are used to:
Examples of polynomial sequences generated by more complex generating functions include: