In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. 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. 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 infinite multi-dimensional arrays of 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 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 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. Also, 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.
Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. 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.
A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.
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
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 that satisfies the linear recurrence relation . The corresponding exponential generating function has the form
and its derivatives can readily be shown to satisfy the differential equation as a direct analogue with the recurrence relation above. In this view, the factorial term is merely a counter-term to normalise the derivative operator acting on .
The Lambert series coefficients in the power series expansions for integers are related by the divisor sum. The main article provides several more classical, or at least well-known examples related to special arithmetic functions in number theory.
In a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.
Examples of generating functions for simple sequencesEdit
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
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 x0 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, 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, ... (which skips over x, x3, x5, ...) 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
The ordinary generating function of a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence with constant coefficients; this generalizes the examples above. Conversely, 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). This observation shows it is easy to solve for generating functions of sequences defined by a linear finite difference equation with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive Binet's formula for the Fibonacci numbers via generating function techniques.
We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate quasi-polynomial sequences of the form 
where the reciprocal roots, , are fixed scalars and where is a polynomial in for all .
In general, Hadamard products of rational functions produce rational generating functions. Similarly, if is a bivariate rational generating function, then its corresponding diagonal generating function, , is algebraic. For example, if we let 
then this generating function's diagonal coefficient generating function is given by the well-known OGF formula
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, ...). See also the section on convolutions in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.
For integers , we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of and , respectively:
Differentiation and integration of generating functionsEdit
We have the following respective power series expansions for the first derivative of a generating function and its integral:
The differentiation–multiplication operation of the second identity can be repeated times to multiply the sequence by , but that requires alternating between differentiation and multiplication. If instead doing differentiations in sequence, the effect is to multiply by the -th falling factorial:
Enumerating arithmetic progressions of sequencesEdit
In this section we give formulas for generating functions enumerating the sequence given an ordinary generating function where , , and (see the main article on transformations). For , this is simply the familiar decomposition of a function into even and odd parts (i.e., even and odd powers):
A formal power series (or function) is said to be holonomic if it satisfies a linear differential equation of the form 
where the coefficients are in the field of rational functions, . Equivalently, is holonomic if the vector space over spanned by the set of all of its derivatives is finite dimensional.
Since we can clear denominators if need be in the previous equation, we may assume that the functions, are polynomials in . Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a P-recurrence of the form
for all large enough and where the are fixed finite-degree polynomials in . In other words, the properties that a sequence be P-recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the Hadamard product operation on generating functions.
The functions , , , , , the dilogarithm function , the generalized hypergeometric functions and the functions defined by the power series and the non-convergent are all holonomic. Examples of P-recursive sequences with holonomic generating functions include and , where sequences such as and are not P-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely-many singularities such as , , and are not holonomic functions.
Software for working with P-recursive sequences and holonomic generating functionsEdit
Tools for processing and working with P-recursive sequences in Mathematica include the software packages provided for non-commercial use on the RISC Combinatorics Group algorithmic combinatorics software site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the Guess package for guessing P-recurrences for arbitrary input sequences (useful for experimental mathematics and exploration) and the Sigma package which is able to find P-recurrences for many sums and solve for closed-form solutions to P-recurrences involving generalized harmonic numbers. Other packages listed on this particular RISC site are targeted at working with holonomic generating functions specifically.
(Depending on how in depth this article gets on the topic, there are many, many other examples of useful software tools that can be listed here or on this page in another section.)
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
where each of A(x) and B(x) is a function that is analytic to a radius of convergence greater than r (or is entire), and where B(r) ≠ 0 then
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,
Bivariate and multivariate generating functionsEdit
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 sequence, in n, and find the generating function in y that has these sequence values as coefficients. Since the generating function for is
the generating function for the binomial coefficients is:
Representation by continued fractions (Jacobi-type J-fractions)Edit
Expansions of (formal) Jacobi-type and Stieltjes-typecontinued fractions (J-fractions and S-fractions, respectively) whose rational convergents represent -order accurate power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the Jacobi-type continued fractions (J-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to for some specific, application-dependent component sequences, and , where denotes the formal variable in the second power series expansion given below:
The coefficients of , denoted in shorthand by , in the previous equations correspond to matrix solutions of the equations
where , for , if , and where for all integers , we have an addition formula relation given by
For (though in practice when ), we can define the rational convergents to the infinite J-fraction, , expanded by
component-wise through the sequences, and , defined recursively by
Moreover, the rationality of the convergent function, for all implies additional finite difference equations and congruence properties satisfied by the sequence of , and for if then we have the congruence
for non-symbolic, determinate choices of the parameter sequences, and , when , i.e., when these sequences do not implicitly depend on an auxiliary parameter such as , , or as in the examples contained in the table below.
The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references )
in several special cases of the prescribed sequences, , generated by the general expansions of the J-fractions defined in the first subsection. Here we define and the parameters , and to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these J-fractions are defined in terms of the q-Pochhammer symbol, Pochhammer symbol, and the binomial coefficients.
The radii of convergence of these series corresponding to the definition of the Jacobi-type J-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.
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
Example 2: Modified binomial coefficient sums and the binomial transformEdit
As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence we define the two sequences of sums
for all , and seek to express the second sums in terms of the first. We suggest an approach by generating functions.
First, we use the binomial transform to write the generating function for the first sum as
Since the generating function for the sequence is given by , we may write the generating function for the second sum defined above in the form
In particular, we may write this modified sum generating function in the form of
for , , , and where .
Finally, it follows that we may express the second sums through the first sums in the following form:
Example 3: Generating functions for mutually recursive sequencesEdit
In this example, we re-formulate a generating function example given in Section 7.3 of Concrete Mathematics (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted ) to tile a rectangle with unmarked domino pieces. Let the auxiliary sequence, , be defined as the number of ways to cover a rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a closed form formula for without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series
If we consider the possible configurations that can be given starting from the left edge of the rectangle, we are able to express the following mutually dependent, or mutually recursive, recurrence relations for our two sequences when defined as above where , , , and :
Since we have that for all integers , the index-shifted generating functions satisfy (incidentally, we also have a corresponding formula when given by ), we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by
which then implies by solving the system of equations (and this is the particular trick to our method here) that
Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that and that
for all integers . We also note that the same shifted generating function technique applied to the second-order recurrence for the Fibonacci numbers is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on rational functions given above.
A discrete convolution of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see Cauchy product).
Consider A(z) and B(z) are ordinary generating functions.
Consider A(z) and B(z) are exponential generating functions.
Consider the triply convolved sequence resulting from the product of three ordinary generating functions
Consider the -fold convolution of a sequence with itself for some positive integer (see the example below for an application)
Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the probability generating function, or pgf, of a random variable is denoted by , then we can show that for any two random variables 
if and are independent. Similarly, the number of ways to pay cents in coin denominations of values in the set (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
and moreover, if we allow the cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the partition function generating function expanded by the infinite q-Pochhammer symbol product of .
Example: The generating function for the Catalan numbersEdit
An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, . In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product so that the order of multiplication is completely specified. For example, which corresponds to the two expressions and . It follows that the sequence satisfies a recurrence relation given by
and so has a corresponding convolved generating function, , satisfying
Since , we then arrive at a formula for this generating function given by
Note that the first equation implicitly defining above implies that
which then leads to another "simple" (as in of form) continued fraction expansion of this generating function.
Example: Spanning trees of fans and convolutions of convolutionsEdit
A fan of order is defined to be a graph on the vertices with edges connected according to the following rules: Vertex is connected by a single edge to each of the other vertices, and vertex is connected by a single edge to the next vertex for all . There is one fan of order one, three fans of order two, eight fans of order three, and so on. A spanning tree is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees of a fan of order are possible for each .
As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when , we have that , which is a sum over the -fold convolutions of the sequence for . More generally, we may write a formula for this sequence as
from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as
from which we are able to extract an exact formula for the sequence by taking the partial fraction expansion of the last generating function.
Implicit generating functions and the Lagrange inversion formulaEdit
This section needs expansion with: This section needs to be added to the list of techniques with generating functions. You can help by adding to it. (April 2017)
Introducing a free parameter (snake oil method)Edit
Sometimes the sum is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.
Both methods discussed so far have as limit in the summation. When n does not appear explicitly in the summation, we may consider as a “free” parameter and treat as a coefficient of , change the order of the summations on and , and try to compute the inner sum.
For example, if we want to compute
we can treat as a "free" parameter, and set
Interchanging summation (“snake oil”) gives
Now the inner sum is . Thus
Then we obtain
It is instructive to use the same method again for the sum, but this time take as the free parameter instead of . We thus set
We say that two generating functions (power series) are congruent modulo , written if their coefficients are congruent modulo for all , i.e., for all relevant cases of the integers (note that we need not assume that is an integer here—it may very well be polynomial-valued in some indeterminate , for example). If the "simpler" right-hand-side generating function, , is a rational function of , then the form of this sequences suggests that the sequence is eventually periodic modulo fixed particular cases of integer-valued . For example, we can prove that the Euler numbers, , satisfy the following congruence modulo :
One of the most useful, if not downright powerful, methods of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by J-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's Lectures on Generating Functions as follows:
Theorem: (Congruences for Series Generated by Expansions of Continued Fractions) — Suppose that the generating function is represented by an infinite continued fraction of the form
and that denotes the convergent to this continued fraction expansion defined such that for all . Then 1) the function is rational for all where we assume that one of divisibility criteria of is met, i.e., for some ; and 2) If the integer divides the product , then we have that .
The main article on the Stirling numbers generated by the finite products
provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference Generatingfunctionology.
We repeat the basic argument and notice that when reduces modulo , these finite product generating functions each satisfy
which implies that the parity of these Stirling numbers matches that of the binomial coefficient
and consequently shows that is even whenever .
Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo to obtain slightly more complicated expressions providing that
In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that thepartition function is generated by the reciprocal infinite q-Pochhammer symbol product (or z-Pochhammer product as the case may be) given by
This partition function satisfies many known congruence properties, which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:
We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.
First, we observe that in the binomial coefficient generating function
all of the coefficients are divisible by 5 except for those which correspond to the powers and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,
It follows that
Using the infinite product expansions of
it can be shown that the coefficient of in is divisible by 5 for all . Finally, since
we may equate the coefficients of in the previous equations to prove our desired congruence result, namely that for all .
There are a number of transformations of generating functions that provide other applications (see the main article). A transformation of a sequence's ordinary generating function (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations).
Generating function transformations can come into play when we seek to express a generating function for the sums
in the form of involving the original sequence generating function. For example, if the sums , then the generating function for the modified sum expressions is given by  (see also the binomial transform and the Stirling transform).
There are also integral formulas for converting between a sequence's OGF, , and its exponential generating function, or EGF, , and vice versa given by
provided that these integrals converge for appropriate values of .
Knuth's article titled "Convolution Polynomials" defines a generalized class of convolution polynomial sequences by their special generating functions of the form
for some analytic function with a power series expansion such that .
We say that a family of polynomials, , forms a convolution family if and if the following convolution condition holds for all and for all :
We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.
A sequence of convolution polynomials defined in the notation above has the following properties:
For arbitrary (fixed) , these polynomials satisfy convolution formulas of the form
For a fixed non-zero parameter , we have modified generating functions for these convolution polynomial sequences given by
where is implicitly defined by a functional equation of the form .
Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, and , with respective corresponding generating functions, and , then for arbitrary we have the identity
An initial listing of special mathematical series is found here. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of Concrete Mathematics and in Section 2.5 of Wilf's Generatingfunctionology. Other special generating functions of note include the entries in the next table, which is by no means complete.
This section needs expansion with: Lists of special and special sequence generating functions. The next table is a start. You can help by adding to it. (April 2017)