Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers ("a""n") 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.
Definitions.
Ordinary generating function (OGF).
The "ordinary generating function" of a sequence "a""n" is
When the term "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 generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array "a""m, n" (where "n" and "m" are natural numbers) is
Exponential generating function (EGF).
The "exponential generating function" of a sequence "a""n" is
Exponential generating functions are generally more convenient than ordinary generating functions for combinatorial enumeration problems that involve labelled objects.
Poisson generating function.
The "Poisson generating function" of a sequence "a""n" is
Lambert series.
The "Lambert series" of a sequence "a""n" is
The Lambert series coefficients in the power series expansions formula_6 for integers formula_7 are related by the divisor sum formula_8. 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.
Bell series.
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
Dirichlet series generating functions (DGFs).
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
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
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 formula_12 if and only if formula_13 where formula_14 is the Riemann zeta function.
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
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.
Ordinary generating functions.
Examples of generating functions for simple sequences.
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 "x"0 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"2, "a"3, ... 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
and the third power has as coefficients the triangular numbers 1, 3, 6, 10, 15, 21, ... whose term "n" is the binomial coefficient formula_21, so that
More generally, for any non-negative integer "k" and non-zero real value "a", it is true 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:
We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the geometric series in the following form:
By induction, we can similarly show for positive integers formula_27 that
where formula_29 denote the Stirling numbers of the second kind and where the generating function formula_30, so that we can form the analogous generating functions over the integral formula_31-th powers generalizing the result in the square case above. In particular, since we can write formula_32, we can apply a well-known finite sum identity involving the Stirling numbers to obtain that
Rational functions.
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, formula_35, are fixed scalars and where formula_36 is a polynomial in formula_37 for all formula_38.
In general, Hadamard products of rational functions produce rational generating functions. Similarly, if formula_39 is a bivariate rational generating function, then its corresponding "diagonal generating function", formula_40, is "algebraic". For example, if we let 
then this generating function's diagonal coefficient generating function is given by the well-known OGF formula
This result is computed in many ways, including Cauchy's integral formula or contour integration, taking complex residues, or by direct manipulations of formal power series in two variables.
Operations on generating functions.
Multiplication yields convolution.
Multiplication of ordinary generating functions yields a discrete convolution (the Cauchy product) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general Euler–Maclaurin 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.
Shifting sequence indices.
For integers formula_27, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of formula_46 and formula_47, respectively:
Differentiation and integration of generating functions.
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 formula_50 times to multiply the sequence by formula_51, but that requires alternating between differentiation and multiplication. If instead doing formula_50 differentiations in sequence, the effect is to multiply by the formula_50th falling factorial:
Using the Stirling numbers of the second kind, that can be turned into another formula for multiplying by formula_51 as follows (see the main article on generating function transformations):
A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the zeta series transformation and its generalizations defined as a derivative-based transformation of generating functions, or alternately termwise by an performing an integral transformation on the sequence generating function. Related operations of performing fractional integration on a sequence generating function are discussed here.
Enumerating arithmetic progressions of sequences.
In this section we give formulas for generating functions enumerating the sequence formula_57 given an ordinary generating function formula_58 where formula_59, formula_60, and formula_61 (see the main article on transformations). For formula_62, this is simply the familiar decomposition of a function into even and odd parts (i.e., even and odd powers):
More generally, suppose that formula_65 and that formula_66 denotes the formula_67th primitive root of unity. Then, as an application of the discrete Fourier transform, we have the formula
For integers formula_27, another useful formula providing somewhat "reversed" floored arithmetic progressions — effectively repeating each coefficient formula_31 times — are generated by the identity
P-recursive sequences and holonomic generating functions.
Definitions.
A formal power series (or function) formula_58 is said to be holonomic if it satisfies a linear differential equation of the form 
where the coefficients formula_74 are in the field of rational functions, formula_75. Equivalently, formula_58 is holonomic if the vector space over formula_75 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, formula_74 are polynomials in formula_79. 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 formula_81 and where the formula_82 are fixed finite-degree polynomials in formula_37. 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 formula_84 on generating functions.
Examples.
The functions formula_85, formula_86, formula_87, formula_88, formula_89, the dilogarithm function formula_90, the generalized hypergeometric functions formula_91 and the functions defined by the power series formula_92 and the non-convergent formula_93 are all holonomic. Examples of P-recursive sequences with holonomic generating functions include formula_94 and formula_95, where sequences such as formula_96 and formula_97 are "not" P-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely-many singularities such as formula_98, formula_99, and formula_100 are "not" holonomic functions.
Software for working with P-recursive sequences and holonomic generating functions.
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. 
Relation to discrete-time Fourier transform.
When the series converges absolutely,
is the discrete-time Fourier transform of the sequence "a"0, "a"1, ….
Asymptotic growth of a sequence.
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"("a""n"; "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
using the Gamma function, a binomial coefficient, or a multiset coefficient.
Often this approach can be iterated to generate several terms in an asymptotic series for "a""n". In particular,
The asymptotic growth of the coefficients of this generating function can then be sought via the finding of "A", "B", "α", "β", and "r" to describe the generating function, as above.
Similar asymptotic analysis is possible for exponential generating functions. With an exponential generating function, it is "a""n"/"n"! that grows according to these asymptotic formulae.
Asymptotic growth of the sequence of squares.
As derived above, the ordinary generating function for the sequence of squares is
With "r" = 1, α = −1, β = 3, "A"("x") = 0, and "B"("x") = "x"+1, we can verify that the squares grow as expected, like the squares:
Asymptotic growth of the Catalan numbers.
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 functions.
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 formula_109 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 formula_110 for all "k" and "n". To do this, consider formula_109 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 formula_112 is
the generating function for the binomial coefficients is:
Representation by continued fractions (Jacobi-type J-fractions).
Definitions.
Expansions of (formal) "Jacobi-type" and "Stieltjes-type" continued fractions ("J-fractions" and "S-fractions", respectively) whose formula_115 rational convergents represent formula_116-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 formula_79 for some specific, application-dependent component sequences, formula_118 and formula_119, where formula_120 denotes the formal variable in the second power series expansion given below:
The coefficients of formula_122, denoted in shorthand by formula_123, in the previous equations correspond to matrix solutions of the equations
where formula_125, formula_126 for formula_7, formula_128 if formula_129, and where for all integers formula_130, we have an "addition formula" relation given by
Properties of the th convergent functions.
For formula_132 (though in practice when formula_133), we can define the rational formula_134 convergents to the infinite J-fraction, formula_135, expanded by
component-wise through the sequences, formula_137 and formula_138, defined recursively by
Moreover, the rationality of the convergent function, formula_140 for all formula_133 implies additional finite difference equations and congruence properties satisfied by the sequence of formula_142, "and" for formula_143 if formula_144 then we have the congruence
for non-symbolic, determinate choices of the parameter sequences, formula_118 and formula_119, when formula_133, i.e., when these sequences do not implicitly depend on an auxiliary parameter such as formula_149, formula_150, or formula_151 as in the examples contained in the table below.
Examples.
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, formula_142, generated by the general expansions of the J-fractions defined in the first subsection. Here we define formula_153 and the parameters formula_151, formula_155 and formula_150 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.
Examples.
Generating functions for the sequence of square numbers "a""n" = "n"2 are:
Lambert series.
As an example of a Lambert series identity not given in the main article, we can show that for formula_159 we have that 
where we have the special case identity for the generating function of the divisor function, formula_161, given by
Dirichlet series generating function.
using the Riemann zeta function.
The sequence "ak" generated by a Dirichlet series generating function (DGF) corresponding to:
where formula_14 is the Riemann zeta function, has the ordinary generating function:
Multivariate generating functions.
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 formula_168 and the column sums are formula_169. Then, according to I. J. Good, the number of such tables is the coefficient of
in
In the bivariate case, non-polynomial double sum examples of so-termed ""double" or "super"" generating functions of the form formula_172 include the following two-variable generating functions for the binomial coefficients, the Stirling numbers, and the Eulerian numbers:
Applications.
Various techniques: Evaluating sums and tackling other problems with generating functions.
Example 1: A formula for sums of harmonic numbers.
Generating functions give us several methods to manipulate sums and to establish identities between sums.
The simplest case occurs when formula_174. We then know that formula_175 for the corresponding ordinary generating functions.
For example, we can manipulate formula_176, where formula_177 are the harmonic numbers. Let formula_178 be the ordinary generating function of the harmonic numbers. Then
and thus
Using formula_181, convolution with the numerator yields
which can also be written as
Example 2: Modified binomial coefficient sums and the binomial transform.
As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence formula_184 we define the two sequences of sums
for all formula_187, 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 formula_189 is given by formula_190, 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 formula_193, formula_194, formula_195, and formula_196 where formula_197.
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 sequences.
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 formula_199) to tile a formula_200 rectangle with unmarked formula_201 domino pieces. Let the auxiliary sequence, formula_202, be defined as the number of ways to cover a formula_200 rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a closed form formula for formula_199 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 formula_200 rectangle, we are able to express the following mutually dependent, or "mutually recursive", recurrence relations for our two sequences when formula_208 defined as above where formula_209, formula_210, formula_211, and formula_212:
Since we have that for all integers formula_214, the index-shifted generating functions satisfy formula_215 (incidentally, we also have a corresponding formula when formula_216 given by formula_217), 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 formula_220 and that
for all integers formula_187. 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.
Convolution (Cauchy products).
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).
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 formula_229 is denoted by formula_230, then we can show that for any two random variables 
if formula_232 and formula_233 are independent. Similarly, the number of ways to pay formula_187 cents in coin denominations of values in the set formula_235 (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product
and moreover, if we allow the formula_37 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 formula_238.
Example: The generating function for the Catalan numbers.
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, formula_239. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product formula_240 so that the order of multiplication is completely specified. For example, formula_241 which corresponds to the two expressions formula_242 and formula_243. It follows that the sequence satisfies a recurrence relation given by
and so has a corresponding convolved generating function, formula_245, satisfying
Since formula_247, we then arrive at a formula for this generating function given by
Note that the first equation implicitly defining formula_245 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 convolutions.
A "fan of order formula_37" is defined to be a graph on the vertices formula_252 with formula_253 edges connected according to the following rules: Vertex formula_254 is connected by a single edge to each of the other formula_37 vertices, and vertex formula_50 is connected by a single edge to the next vertex formula_257 for all formula_258. 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 formula_259 of a fan of order formula_37 are possible for each formula_7.
As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when formula_262, we have that formula_263, which is a sum over the formula_31-fold convolutions of the sequence formula_265 for formula_266. 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.
Introducing a free parameter (snake oil method).
Sometimes the sum formula_269 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 formula_37 as limit in the summation. When n does not appear explicitly in the summation, we may consider formula_37 as a “free” parameter and treat formula_269 as a coefficient of formula_273, change the order of the summations on formula_37 and formula_50, and try to compute the inner sum.
For example, if we want to compute
we can treat formula_37 as a "free" parameter, and set
Interchanging summation (“snake oil”) gives
Now the inner sum is formula_280. Thus
Then we obtain
Generating functions prove congruences.
We say that two generating functions (power series) are congruent modulo formula_31, written formula_284 if their coefficients are congruent modulo formula_31 for all formula_187, i.e., formula_287 for all relevant cases of the integers formula_37 (note that we need not assume that formula_31 is an integer here—it may very well be polynomial-valued in some indeterminate formula_150, for example). If the ""simpler" right-hand-side generating function, formula_291, is a rational function of formula_79, then the form of this sequences suggests that the sequence is eventually periodic modulo fixed particular cases of integer-valued formula_293. For example, we can prove that the Euler numbers, formula_294, satisfy the following congruence modulo formula_295:
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" formula_297) 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:
Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the Stirling numbers of the first kind and for the partition function (mathematics) formula_312 which show the versatility of generating functions in tackling problems involving integer sequences.
The Stirling numbers modulo small integers.
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 formula_314, 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 formula_317 is even whenever formula_318.
Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo formula_295 to obtain slightly more complicated expressions providing that
Congruences for the partition function.
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 "the" partition function formula_312 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 the binomial coefficient generating function, formula_324, satisfies that each of its coefficients are divisible by formula_325 with the exception of those which correspond to the powers of formula_326, all of which otherwise have a remainder of formula_327 modulo formula_325. Thus we may write
which in particular shows us that
Hence, we easily see that formula_325 divides each coefficient of formula_332 in the infinite product expansions of
Finally, since we may write the generating function for the partition function as
we may equate the coefficients of formula_335 in the previous equations to prove our desired congruence result, namely that, formula_336 for all formula_214.
Transformations of generating functions.
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 formula_339 involving the original sequence generating function. For example, if the sums formula_340, then the generating function for the modified sum expressions is given by formula_341 (see also the binomial transform and the Stirling transform).
 exercise 5.71
There are also integral formulas for converting between a sequence's OGF, formula_58, and its exponential generating function, or EGF, formula_343, and vice versa given by
provided that these integrals converge for appropriate values of formula_79.
Other applications.
Generating functions are used to:
Other generating functions.
Examples.
Examples of polynomial sequences generated by more complex generating functions include:
Other sequences generated by more complex generating functions:
Convolution polynomials.
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 formula_348 with a power series expansion such that formula_349. 
We say that a family of polynomials, formula_350, forms a "convolution family" if formula_351 and if the following convolution condition holds for all formula_352 and for all formula_187:
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 a fixed non-zero parameter formula_360, we have modified generating functions for these convolution polynomial sequences given by
where formula_362 is implicitly defined by a functional equation of the form formula_363. 
Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, formula_364 and formula_365, with respective corresponding generating functions, formula_366 and formula_367, then for arbitrary formula_368 we have the identity
Examples of convolution polynomial sequences include the "binomial power series", formula_370, so-termed "tree polynomials", the Bell numbers, formula_371, the Laguerre polynomials, and the Stirling convolution polynomials.
Tables of special generating functions.
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.
! Formal power series !! Generating-function formula !! Notes