Generating function

In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.

There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. 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 sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.

Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.

George Pólya writes in Mathematics and plausible reasoning:

, George Pólya, Mathematics and plausible reasoning (1954)

, Herbert Wilf, Generatingfunctionology (1994)

Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. Thus generating functions are not functions in the formal sense of a mapping from a domain to a codomain.

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 series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x.

Not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.

When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function. The ordinary generating function of a sequence a_n is: \[G(a_n;x)=\sum_{n=0}^\infty a_n x^n.\] 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 exponential generating function of a sequence a_n is \[\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.\]

Exponential generating functions are generally more convenient than ordinary generating functions for combinatorial enumeration problems that involve labelled objects.

Another benefit of exponential generating functions is that they are useful in transferring linear recurrence relations to the realm of differential equations. For example, take the Fibonacci sequence {f_n} that satisfies the linear recurrence relation f_n+2 = f_n+1 + f_n. The corresponding exponential generating function has the form \[\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n\]

and its derivatives can readily be shown to satisfy the differential equation EF″(x) = EF′(x) + EF(x) as a direct analogue with the recurrence relation above. In this view, the factorial term n! is merely a counter-term to normalise the derivative operator acting on x.

The Poisson generating function of a sequence a_n is \[\operatorname{PG}(a_n;x)=\sum _{n=0}^\infty a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).\]

The Lambert series of a sequence a_n is \[\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.\]Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.

The Lambert series coefficients in the power series expansions \[b_n := [x^n] \operatorname{LG}(a_n;x)\]for integers n ≥ 1 are related by the divisor sum \[b_n = \sum_{d|n} a_d.\]The main article provides several more classical, or at least well-known examples related to special arithmetic functions in number theory. As an example of a Lambert series identity not given in the main article, we can show that for |x|, |xq| < 1 we have that \[\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n},\]

where we have the special case identity for the generating function of the divisor function, d(n) ≡ σ₀(n), given by\[\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}.\]

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: \[\operatorname{BG}_p(a_n;x) = \sum_{n=0}^\infty a_{p^n}x^n.\]

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_n is: \[\operatorname{DG}(a_n;s)=\sum _{n=1}^\infty \frac{a_n}{n^s}.\]

The Dirichlet series generating function is especially useful when a_n is a multiplicative function, in which case it has an Euler product expression in terms of the function's Bell series: \[\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.\]

If a_n is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series. We also have a relation between the pair of coefficients in the Lambert series expansions above and their DGFs. Namely, we can prove that: \[[x^n] \operatorname{LG}(a_n; x) = b_n\]if and only if \[\operatorname{DG}(a_n;s) \zeta(s) = \operatorname{DG}(b_n;s),\]where ζ(s) is the Riemann zeta function.

The sequence a_k generated by a Dirichlet series generating function (DGF) corresponding to:\[\operatorname{DG}(a_k;s)=\zeta(s)^m\]has the ordinary generating function:\[\sum_{k=1}^{k=n} a_k x^k = x + \binom{m}{1} \sum_{2 \leq a \leq n} x^{a} + \binom{m}{2}\underset{ab \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty} x^{ab} + \binom{m}{3}\underset{abc \leq n}{\sum_{a = 2}^\infty \sum_{c = 2}^\infty \sum_{b = 2}^\infty} x^{abc} + \binom{m}{4}\underset{abcd \leq n}{\sum_{a = 2}^\infty \sum_{b = 2}^\infty \sum_{c = 2}^\infty \sum_{d = 2}^\infty} x^{abcd} + \cdots\]

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by: \[e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n\]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 of polynomial sequences generated by more complex generating functions include:

• Appell polynomials
• Chebyshev polynomials
• Difference polynomials
• Generalized Appell polynomials
• q-difference polynomials

Other sequences generated by more complex generating functions include:

• Double exponential generating functions
• Hadamard products of generating functions and diagonal generating functions, and their corresponding integral transformations

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 fundamental generating function is that of the constant sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., whose ordinary generating function is the geometric series \[\sum_{n=0}^\infty x^n= \frac{1}{1-x}.\]

The left-hand side is the Maclaurin series expansion of the right-hand side. Alternatively, the equality 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 are equal to 0). 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, a, a, ... for any constant a: \[\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.\]

(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, \[\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.\]

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, ... (which skips over x, x, x, ...) one gets the generating function \[\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.\]

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 \[\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},\]

Condensed: the full section is in Wikipedia.