Editing Generating function (section)

== Ordinary generating functions ==

=== Examples 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 {{nowrap|1, 1, 1, 1, 1, 1, 1, 1, 1, ...}}, whose ordinary generating function is the [[Geometric_series#Closed-form_formula|geometric series]]
<math display="block">\sum_{n=0}^\infty x^n= \frac{1}{1-x}.</math>

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 {{math|1 − ''x''}}, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of {{math|''x''<sup>0</sup>}} 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 {{math|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 {{math|''x'' → ''ax''}} gives the generating function for the [[Geometric progression|geometric sequence]] {{math|1, ''a'', ''a''<sup>2</sup>, ''a''<sup>3</sup>, ...}} for any constant {{mvar|a}}:
<math display="block">\sum_{n=0}^\infty(ax)^n= \frac{1}{1-ax}.</math>

(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,
<math display="block">\sum_{n=0}^\infty(-1)^nx^n= \frac{1}{1+x}.</math>

One can also introduce regular gaps in the sequence by replacing {{mvar|x}} by some power of {{mvar|x}}, so for instance for the sequence {{nowrap|1, 0, 1, 0, 1, 0, 1, 0, ...}} (which skips over {{math|''x'', ''x''<sup>3</sup>, ''x''<sup>5</sup>, ...}}) one gets the generating function
<math display="block">\sum_{n=0}^\infty x^{2n} = \frac{1}{1-x^2}.</math>

By squaring the initial generating function, or by finding the derivative of both sides with respect to {{mvar|x}} and making a change of running variable {{math|''n'' → ''n'' + 1}}, one sees that the coefficients form the sequence {{nowrap|1, 2, 3, 4, 5, ...}}, so one has
<math display="block">\sum_{n=0}^\infty(n+1)x^n= \frac{1}{(1-x)^2},</math>

and the third power has as coefficients the [[triangular number]]s {{nowrap|1, 3, 6, 10, 15, 21, ...}} whose term {{mvar|n}} is the [[binomial coefficient]] {{math|{{pars|s=150%|{{su|p=''n'' + 2|b=2|a=c}}}}}}, so that
<math display="block">\sum_{n=0}^\infty\binom{n+2}2 x^n= \frac{1}{(1-x)^3}.</math>

More generally, for any non-negative integer {{mvar|k}} and non-zero real value {{mvar|a}}, it is true that
<math display="block">\sum_{n=0}^\infty a^n\binom{n+k}k x^n= \frac{1}{(1-ax)^{k+1}}\,.</math>

Since
<math display="block">2\binom{n+2}2 - 3\binom{n+1}1 + \binom{n}0 = 2\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2,</math>

one can find the ordinary generating function for the sequence {{nowrap|0, 1, 4, 9, 16, ...}} of [[square number]]s by linear combination of binomial-coefficient generating sequences:
<math display="block">G(n^2;x) = \sum_{n=0}^\infty n^2x^n = \frac{2}{(1-x)^3} - \frac{3}{(1-x)^2} + \frac{1}{1-x} = \frac{x(x+1)}{(1-x)^3}.</math>

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:
<math display="block">\begin{align}
G(n^2;x)
 & = \sum_{n=0}^\infty n^2x^n \\[4px]
 & = \sum_{n=0}^\infty n(n-1) x^n + \sum_{n=0}^\infty n x^n \\[4px]
 & = x^2 D^2\left[\frac{1}{1-x}\right] + x D\left[\frac{1}{1-x}\right] \\[4px]
 & = \frac{2 x^2}{(1-x)^3} + \frac{x}{(1-x)^2} =\frac{x(x+1)}{(1-x)^3}.
\end{align}</math>

By induction, we can similarly show for positive integers {{math|''m'' ≥ 1}} that<ref>{{cite journal|first1= Michael Z. | last1=Spivey | title=Combinatorial Sums and Finite Differences | year=2007 |journal = Discrete Math. |doi = 10.1016/j.disc.2007.03.052 | volume=307|number=24|pages=3130–3146|mr=2370116|doi-access=free }}</ref><ref>{{cite arXiv|first1=R. J. |last1=Mathar|year=2012|eprint=1207.5845|title=Yet another table of integrals|class=math.CA}} v4 eq. (0.4)</ref>
<math display="block">n^m = \sum_{j=0}^m \begin{Bmatrix} m \\ j \end{Bmatrix} \frac{n!}{(n-j)!}, </math>

where {{math|{{resize|150%|{}}{{su|p=''n''|b=''k''}}{{resize|150%|}<nowiki/>}}}} denote the [[Stirling numbers of the second kind]] and where the generating function
<math display="block">\sum_{n = 0}^\infty \frac{n!}{ (n-j)!} \, z^n = \frac{j! \cdot z^j}{(1-z)^{j+1}},</math>

so that we can form the analogous generating functions over the integral {{mvar|m}}th powers generalizing the result in the square case above. In particular, since we can write
<math display="block">\frac{z^k}{(1-z)^{k+1}} = \sum_{i=0}^k \binom{k}{i} \frac{(-1)^{k-i}}{(1-z)^{i+1}},</math>

we can apply a well-known finite sum identity involving the [[Stirling numbers]] to obtain that<ref>{{harvnb|Graham|Knuth|Patashnik|1994|loc=Table 265 in §6.1}} for finite sum identities involving the Stirling number triangles.</ref>
<math display="block">\sum_{n = 0}^\infty n^m z^n = \sum_{j=0}^m \begin{Bmatrix} m+1 \\ j+1 \end{Bmatrix} \frac{(-1)^{m-j} j!}{(1-z)^{j+1}}. </math>

=== Rational functions ===
{{Main|Linear recursive sequence}}

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 <ref name="GFLECT">{{harvnb|Lando|2003|loc=§2.4}}</ref>
<math display="block">f_n = p_1(n) \rho_1^n + \cdots + p_\ell(n) \rho_\ell^n, </math>

where the reciprocal roots, <math>\rho_i \isin \mathbb{C}</math>, are fixed scalars and where {{math|''p''<sub>''i''</sub>(''n'')}} is a polynomial in {{mvar|n}} for all {{math|1 ≤ ''i'' ≤ ''ℓ''}}.

In general, [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard products]] of rational functions produce rational generating functions. Similarly, if
<math display="block">F(s, t) := \sum_{m,n \geq 0} f(m, n) w^m z^n</math>

is a bivariate rational generating function, then its corresponding ''diagonal generating function'',
<math display="block">\operatorname{diag}(F) := \sum_{n = 0}^\infty f(n, n) z^n,</math>

is ''algebraic''. For example, if we let<ref>Example from {{cite book |chapter=§6.3 |first1=Richard P. |last1=Stanley |first2=Sergey |last2=Fomin |title=Enumerative Combinatorics: Volume 2 |url=https://books.google.com/books?id=zg5wDqT6T-UC |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-78987-5 |series=Cambridge Studies in Advanced Mathematics |volume=62}}</ref>
<math display="block">F(s, t) := \sum_{i,j \geq 0} \binom{i+j}{i} s^i t^j = \frac{1}{1-s-t}, </math>

then this generating function's diagonal coefficient generating function is given by the well-known OGF formula
<math display="block">\operatorname{diag}(F) = \sum_{n = 0}^\infty \binom{2n}{n} z^n = \frac{1}{\sqrt{1-4z}}. </math>

This result is computed in many ways, including [[Cauchy's integral formula]] or [[contour integration]], taking complex [[residue (complex analysis)|residue]]s, or by direct manipulations of [[formal power series]] in two variables.

=== Operations on generating functions ===

==== Multiplication yields convolution ====
{{Main|Cauchy product}}

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]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''a<sub>n</sub>''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[#Convolution (Cauchy products)|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 {{math|''m'' ≥ 1}}, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of {{math|⟨ ''g''<sub>''n'' − ''m''</sub> ⟩}} and {{math|⟨ ''g''<sub>''n'' + ''m''</sub> ⟩}}, respectively:
<math display="block">\begin{align}
& z^m G(z) = \sum_{n = m}^\infty g_{n-m} z^n \\[4px]
& \frac{G(z) - g_0 - g_1 z - \cdots - g_{m-1} z^{m-1}}{z^m} = \sum_{n = 0}^\infty g_{n+m} z^n.
\end{align}</math>

==== 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:
<math display="block">\begin{align}
G'(z) & = \sum_{n = 0}^\infty (n+1) g_{n+1} z^n \\[4px]
z \cdot G'(z) & = \sum_{n = 0}^\infty n g_{n} z^n \\[4px]
\int_0^z G(t) \, dt & = \sum_{n = 1}^\infty \frac{g_{n-1}}{n} z^n.
\end{align}</math>

The differentiation–multiplication operation of the second identity can be repeated {{mvar|k}} times to multiply the sequence by {{math|''n''<sup>''k''</sup>}}, but that requires alternating between differentiation and multiplication. If instead doing {{mvar|k}} differentiations in sequence, the effect is to multiply by the {{mvar|k}}th [[falling factorial]]:
<math display="block"> z^k G^{(k)}(z) = \sum_{n = 0}^\infty n^\underline{k} g_n z^n = \sum_{n = 0}^\infty n (n-1) \dotsb (n-k+1) g_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

Using the [[Stirling numbers of the second kind]], that can be turned into another formula for multiplying by <math>n^k</math> as follows (see the main article on [[Generating function transformation#Derivative transformations|generating function transformations]]):
<math display="block"> \sum_{j=0}^k \begin{Bmatrix} k \\ j \end{Bmatrix} z^j F^{(j)}(z) = \sum_{n = 0}^\infty n^k f_n z^n \quad\text{for all } k \in \mathbb{N}. </math>

A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the [[Generating function transformation#Derivative transformations|zeta series transformation]] and its generalizations defined as a derivative-based [[generating function transformation|transformation of generating functions]], or alternately termwise by and performing an [[Generating function transformation#Polylogarithm series transformations|integral transformation]] on the sequence generating function. Related operations of performing [[fractional calculus|fractional integration]] on a sequence generating function are discussed [[Generating function transformation#Fractional integrals and derivatives|here]].

==== Enumerating arithmetic progressions of sequences ====
In this section we give formulas for generating functions enumerating the sequence {{math|{''f''<sub>''an'' + ''b''</sub>}<nowiki/>}} given an ordinary generating function {{math|''F''(''z'')}}, where {{math|''a'' ≥ 2}}, {{math|0 ≤ ''b'' < ''a''}}, and {{math|''a''}} and {{math|''b''}} are integers (see the [[generating function transformation|main article on transformations]]). For {{math|''a'' {{=}} 2}}, this is simply the familiar decomposition of a function into [[even and odd functions|even and odd parts]] (i.e., even and odd powers):
<math display="block">\begin{align}
\sum_{n = 0}^\infty f_{2n} z^{2n} &= \frac{F(z) + F(-z)}{2} \\[4px]
\sum_{n = 0}^\infty f_{2n+1} z^{2n+1} &= \frac{F(z) - F(-z)}{2}.
\end{align}</math>

More generally, suppose that {{math|''a'' ≥ 3}} and that {{math|''ω<sub>a</sub>'' {{=}} exp {{sfrac|2''πi''|''a''}}}} denotes the {{mvar|a}}th [[root of unity|primitive root of unity]]. Then, as an application of the [[discrete Fourier transform]], we have the formula<ref name="TAOCPV1">{{harvnb|Knuth|1997|loc=§1.2.9}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{an+b} z^{an+b} = \frac{1}{a} \sum_{m=0}^{a-1} \omega_a^{-mb} F\left(\omega_a^m z\right).</math>

For integers {{math|''m'' ≥ 1}}, another useful formula providing somewhat ''reversed'' floored arithmetic progressions — effectively repeating each coefficient {{mvar|m}} times — are generated by the identity<ref>Solution to {{harvnb|Graham|Knuth|Patashnik|1994|p=569, exercise 7.36}}</ref>
<math display="block">\sum_{n = 0}^\infty f_{\left\lfloor \frac{n}{m} \right\rfloor} z^n = \frac{1-z^m}{1-z} F(z^m) = \left(1 + z + \cdots + z^{m-2} + z^{m-1}\right) F(z^m).</math>

==={{math|''P''}}-recursive sequences and holonomic generating functions===

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>
<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''c<sub>i</sub>''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> 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, {{math|''c<sub>i</sub>''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form
<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''<sub>0</sub>}} and where the {{math|''ĉ<sub>i</sub>''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.

====Examples====

The functions {{math|''e''<sup>''z''</sup>}}, {{math|log ''z''}}, {{math|cos ''z''}}, {{math|arcsin ''z''}}, <math>\sqrt{1 + z}</math>, the [[dilogarithm]] function {{math|Li<sub>2</sub>(''z'')}}, the [[generalized hypergeometric function]]s {{math|''<sub>p</sub>F<sub>q</sub>''(...; ...; ''z'')}} and the functions defined by the power series
<math display="block">\sum_{n = 0}^\infty \frac{z^n}{(n!)^2}</math>

and the non-convergent
<math display="block">
\sum_{n = 0}^\infty n! \cdot z^n
</math>
are all holonomic.

Examples of {{mvar|P}}-recursive sequences with holonomic generating functions include {{math|''f''<sub>''n''</sub> ≔ {{sfrac|1|''n'' + 1}} {{pars|s=150%|{{su|p=2''n''|b=''n''|a=c}}}}}} and {{math|''f''<sub>''n''</sub> ≔ {{sfrac|2<sup>''n''</sup>|''n''<sup>2</sup> + 1}}}}, where sequences such as <math>\sqrt{n}</math> and {{math|log ''n''}} are ''not'' {{mvar|P}}-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as {{math|tan ''z''}}, {{math|sec ''z''}}, and [[Gamma function|{{math|Γ(''z'')}}]] are ''not'' holonomic functions.

====Software for working with ''{{mvar|P}}''-recursive sequences and holonomic generating functions====

Tools for processing and working with {{mvar|P}}-recursive sequences in ''[[Mathematica]]'' include the software packages provided for non-commercial use on the [https://www.risc.jku.at/research/combinat/software/ RISC Combinatorics Group algorithmic combinatorics software] site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>'''Guess'''</code> package for guessing ''{{mvar|P}}-recurrences'' for arbitrary input sequences (useful for [[experimental mathematics]] and exploration) and the <code>'''Sigma'''</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to {{mvar|P}}-recurrences involving generalized [[harmonic number]]s.<ref>{{cite journal|last1=Schneider|first1=C.|title=Symbolic Summation Assists Combinatorics|journal=Sém. Lothar. Combin.|date=2007|volume=56|pages=1–36 |url=http://www.emis.de/journals/SLC/wpapers/s56schneider.html}}</ref> 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, or most appropriately, on a dedicated webpage of its own.-->

=== Relation to discrete-time Fourier transform ===
{{Main|Discrete-time Fourier transform}}

When the series [[Absolute convergence|converges absolutely]],
<math display="block">G \left ( a_n; e^{-i \omega} \right) = \sum_{n=0}^\infty a_n e^{-i \omega n}</math>
is the discrete-time Fourier transform of the sequence {{math|''a''<sub>0</sub>, ''a''<sub>1</sub>, ...}}.

=== 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 analysis|asymptotic growth]] of the underlying sequence.

For instance, if an ordinary generating function {{math|''G''(''a''<sub>''n''</sub>; ''x'')}} that has a finite radius of convergence of {{mvar|r}} can be written as
<math display="block">G(a_n; x) = \frac{A(x) + B(x) \left (1- \frac{x}{r} \right )^{-\beta}}{x^\alpha}</math>

where each of {{math|''A''(''x'')}} and {{math|''B''(''x'')}} is a function that is [[analytic function|analytic]] to a radius of convergence greater than {{mvar|r}} (or is [[Entire function|entire]]), and where {{math|''B''(''r'') ≠ 0}} then
<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1}\left(\frac{1}{r}\right)^n \sim \frac{B(r)}{r^{\alpha}} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n = \frac{B(r)}{r^\alpha} \left(\!\!\binom{\beta}{n}\!\!\right)\left(\frac{1}{r}\right)^n\,,</math>
using the [[gamma function]], a [[binomial coefficient]], or a [[multiset coefficient]].  Note that limit as {{mvar|n}} goes to infinity of the ratio of {{math|''a''{{sub|''n''}}}} to any of these expressions is guaranteed to be 1; not merely that {{math|''a''{{sub|''n''}}}} is proportional to them.

Often this approach can be iterated to generate several terms in an asymptotic series for {{math|''a''<sub>''n''</sub>}}. In particular,
<math display="block">G\left(a_n - \frac{B(r)}{r^\alpha} \binom{n+\beta-1}{n}\left(\frac{1}{r}\right)^n; x \right) = G(a_n; x) - \frac{B(r)}{r^\alpha} \left(1 - \frac{x}{r}\right)^{-\beta}\,.</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of {{mvar|A}}, {{mvar|B}}, {{mvar|α}}, {{mvar|β}}, and {{mvar|r}} to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is {{math|{{sfrac|''a''<sub>''n''</sub>|''n''!}}}} that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

==== Asymptotic growth of the sequence of squares ====
As derived above, the ordinary generating function for the sequence of squares is:
<math display="block">G(n^2; x) = \frac{x(x+1)}{(1-x)^3}.</math>

With {{math|1=''r'' = 1}}, {{math|1=''α'' = −1}}, {{math|1=''β'' = 3}}, {{math|1=''A''(''x'') = 0}}, and {{math|1=''B''(''x'') = ''x'' + 1}}, we can verify that the squares grow as expected, like the squares:
<math display="block">a_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left (\frac{1}{r} \right)^n = \frac{1+1}{1^{-1}\,\Gamma(3)}\,n^{3-1} \left(\frac1 1\right)^n = n^2.</math>

==== Asymptotic growth of the Catalan numbers ====
{{Main|Catalan number}}

The ordinary generating function for the [[Catalan number]]s is
<math display="block">G(C_n; x) = \frac{1-\sqrt{1-4x}}{2x}.</math>

With {{math|1=''r'' = {{sfrac|1|4}}}}, {{math|1=''α'' = 1}}, {{math|1=''β'' = −{{sfrac|1|2}}}}, {{math|1=''A''(''x'') = {{sfrac|1|2}}}}, and {{math|1=''B''(''x'') = −{{sfrac|1|2}}}}, we can conclude that, for the Catalan numbers:
<math display="block">C_n \sim \frac{B(r)}{r^\alpha \Gamma(\beta)} \, n^{\beta-1} \left(\frac{1}{r} \right)^n = \frac{-\frac12}{\left(\frac14\right)^1 \Gamma\left(-\frac12\right)} \, n^{-\frac12-1} \left(\frac{1}{\,\frac14\,}\right)^n = \frac{4^n}{n^\frac32 \sqrt\pi}.</math>

=== Bivariate and multivariate generating functions ===
The generating function in several variables can be generalized to arrays with multiple indices. These non-polynomial double sum examples are called '''multivariate generating functions''', or '''super generating functions'''. For two variables, these are often called '''bivariate generating functions'''.

==== Bivariate case ====
The ordinary generating function of a two-dimensional array {{math|''a''<sub>''m'',''n''</sub>}} (where {{mvar|n}} and {{mvar|m}} are natural numbers) is:
<math display="block">G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n} x^m y^n.</math>For instance, since {{math|(1 + ''x'')<sup>''n''</sup>}} is the ordinary generating function for [[binomial coefficients]] for a fixed {{mvar|n}}, one may ask for a bivariate generating function that generates the binomial coefficients {{math|{{pars|s=150%|{{su|p=''n''|b=''k''|a=c}}}}}} for all {{mvar|k}} and {{mvar|n}}. To do this, consider {{math|(1 + ''x'')<sup>''n''</sup>}} itself as a sequence in {{mvar|n}}, and find the generating function in {{mvar|y}} that has these sequence values as coefficients. Since the generating function for {{math|''a''<sup>''n''</sup>}} is:
<math display="block">\frac{1}{1-ay},</math>the generating function for the binomial coefficients is:
<math display="block">\sum_{n,k} \binom{n}{k} x^k y^n = \frac{1}{1-(1+x)y}=\frac{1}{1-y-xy}.</math>Other examples of such include the following two-variable generating functions for the [[binomial coefficients]], the [[Stirling numbers]], and the [[Eulerian numbers]], where {{math|''&omega;''}} and {{math|''z''}} denote the two variables:<ref>See the usage of these terms in {{harvnb|Graham|Knuth|Patashnik|1994|loc=§7.4}} on special sequence generating functions.</ref>
<math display="block">\begin{align}
e^{z+wz} & = \sum_{m,n \geq 0} \binom{n}{m} w^m \frac{z^n}{n!} \\[4px]
e^{w(e^z-1)} & = \sum_{m,n \geq 0} \begin{Bmatrix} n \\ m \end{Bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1}{(1-z)^w} & = \sum_{m,n \geq 0} \begin{bmatrix} n \\ m \end{bmatrix} w^m \frac{z^n}{n!} \\[4px]
\frac{1-w}{e^{(w-1)z}-w} & = \sum_{m,n \geq 0} \left\langle\begin{matrix} n \\ m \end{matrix} \right\rangle w^m \frac{z^n}{n!} \\[4px]
\frac{e^w-e^z}{w e^z-z e^w} &= \sum_{m,n \geq 0} \left\langle\begin{matrix} m+n+1 \\ m \end{matrix} \right\rangle \frac{w^m z^n}{(m+n+1)!}.
\end{align}</math>

==== Multivariate case ====
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 {{mvar|r}} rows and {{mvar|c}} columns; the row sums are {{math|''t''<sub>1</sub>, ''t''<sub>2</sub> ... ''t<sub>r</sub>''}} and the column sums are {{math|''s''<sub>1</sub>, ''s''<sub>2</sub> ... ''s<sub>c</sub>''}}. Then, according to [[I. J. Good]],<ref name="Good 1986">{{cite journal |last=Good |first=I. J. |year=1986 |title=On applications of symmetric Dirichlet distributions and their mixtures to contingency tables |journal=[[Annals of Statistics]] |volume=4 |issue=6 |pages=1159–1189 |doi=10.1214/aos/1176343649 |doi-access=free}}</ref> the number of such tables is the coefficient of:
<math display="block">x_1^{t_1}\cdots x_r^{t_r}y_1^{s_1}\cdots y_c^{s_c}</math>in:<math display="block">\prod_{i=1}^{r}\prod_{j=1}^c\frac{1}{1-x_iy_j}.</math>

=== Representation by continued fractions (Jacobi-type ''{{mvar|J}}''-fractions) ===

==== Definitions ====

Expansions of (formal) ''Jacobi-type'' and ''Stieltjes-type'' [[generalized continued fraction|continued fractions]] (''{{mvar|J}}-fractions'' and ''{{mvar|S}}-fractions'', respectively) whose {{mvar|h}}th rational convergents represent [[Order of accuracy|{{math|2''h''}}-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 fraction]]s ({{mvar|J}}-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to {{mvar|z}} for some specific, application-dependent component sequences, {{math|{ab<sub>''i''</sub>}<nowiki/>}} and {{math|{''c''<sub>''i''</sub>}<nowiki/>}}, where {{math|''z'' ≠ 0}} denotes the formal variable in the second power series expansion given below:<ref>For more complete information on the properties of {{mvar|J}}-fractions see:
*{{cite journal |first=P. |last=Flajolet |title=Combinatorial aspects of continued fractions |journal=Discrete Mathematics |volume=32 |issue=2 |pages=125–161 |year=1980 |doi=10.1016/0012-365X(80)90050-3 |url=http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf}}
*{{cite book |first=H.S. |last=Wall |title=Analytic Theory of Continued Fractions |url=https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7 |date=2018 |orig-year=1948 |publisher=Dover |isbn=978-0-486-83044-5}}</ref>
<math display="block">\begin{align}
J^{[\infty]}(z) & = \cfrac{1}{1-c_1 z-\cfrac{\text{ab}_2 z^2}{1-c_2 z-\cfrac{\text{ab}_3 z^2}{\ddots}}} \\[4px]
 & = 1 + c_1 z + \left(\text{ab}_2+c_1^2\right) z^2 + \left(2 \text{ab}_2 c_1+c_1^3 + \text{ab}_2 c_2\right) z^3 + \cdots
\end{align}</math>

The coefficients of <math>z^n</math>, denoted in shorthand by {{math|''j<sub>n</sub>'' ≔ [''z<sup>n</sup>''] ''J''<sup>[∞]</sup>(''z'')}}, in the previous equations correspond to matrix solutions of the equations:
<math display="block">\begin{bmatrix}k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ k_{0,3} & k_{1,3} & k_{2,3} & k_{3,3} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} =
 \begin{bmatrix}k_{0,0} & 0 & 0 & 0 & \cdots \\ k_{0,1} & k_{1,1} & 0 & 0 & \cdots \\ k_{0,2} & k_{1,2} & k_{2,2} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} \cdot
 \begin{bmatrix}c_1 & 1 & 0 & 0 & \cdots \\ \text{ab}_2 & c_2 & 1 & 0 & \cdots \\ 0 & \text{ab}_3 & c_3 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix},
</math>

where {{math|''j''<sub>0</sub> ≡ ''k''<sub>0,0</sub> {{=}} 1}}, {{math|''j<sub>n</sub>'' {{=}} ''k''<sub>0,''n''</sub>}} for {{math|''n'' ≥ 1}}, {{math|''k''<sub>''r'',''s''</sub> {{=}} 0}} if {{math|''r'' > ''s''}}, and where for all integers {{math|''p'', ''q'' ≥ 0}}, we have an ''addition formula'' relation given by:
<math display="block">j_{p+q} = k_{0,p} \cdot k_{0,q} + \sum_{i=1}^{\min(p, q)} \text{ab}_2 \cdots \text{ab}_{i+1} \times k_{i,p} \cdot k_{i,q}. </math>

==== Properties of the ''{{mvar|h}}''th convergent functions ====

For {{math|''h'' ≥ 0}} (though in practice when {{math|''h'' ≥ 2}}), we can define the rational {{mvar|h}}th convergents to the infinite {{mvar|J}}-fraction, {{math|''J''<sup>[∞]</sup>(''z'')}}, expanded by:
<math display="block">\operatorname{Conv}_h(z) := \frac{P_h(z)}{Q_h(z)} = j_0 + j_1 z + \cdots + j_{2h-1} z^{2h-1} + \sum_{n = 2h}^\infty \widetilde{j}_{h,n} z^n</math>

component-wise through the sequences, {{math|''P<sub>h</sub>''(''z'')}} and {{math|''Q<sub>h</sub>''(''z'')}}, defined recursively by:
<math display="block">\begin{align}
P_h(z) & = (1-c_h z) P_{h-1}(z) - \text{ab}_h z^2 P_{h-2}(z) + \delta_{h,1} \\
Q_h(z) & = (1-c_h z) Q_{h-1}(z) - \text{ab}_h z^2 Q_{h-2}(z) + (1-c_1 z) \delta_{h,1} + \delta_{0,1}.
\end{align}</math>

Moreover, the rationality of the convergent function {{math|Conv<sub>''h''</sub>(''z'')}} for all {{math|''h'' ≥ 2}} implies additional finite difference equations and congruence properties satisfied by the sequence of {{math|''j<sub>n</sub>''}}, ''and'' for {{math|''M<sub>h</sub>'' ≔ ab<sub>2</sub> ⋯ ab<sub>''h'' + 1</sub>}} if {{math|''h'' ‖  ''M''<sub>''h''</sub>}} then we have the congruence
<math display="block">j_n \equiv [z^n] \operatorname{Conv}_h(z) \pmod h, </math>

for non-symbolic, determinate choices of the parameter sequences {{math|{ab<sub>''i''</sub>}<nowiki/>}} and {{math|{''c''<sub>''i''</sub>}<nowiki/>}} when {{math|''h'' ≥ 2}}, that is, when these sequences do not implicitly depend on an auxiliary parameter such as {{mvar|q}}, {{mvar|x}}, or {{mvar|R}} 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<ref>See the following articles:
*{{cite arXiv |first=Maxie D. |last=Schmidt |eprint=1612.02778 |title=Continued Fractions for Square Series Generating Functions |year=2016 |class=math.NT }}
*{{cite journal |author-mask= 1 |first=Maxie D. |last=Schmidt |title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions |journal=Journal of Integer Sequences |volume=20 |id=17.3.4 |year=2017 |arxiv=1610.09691 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html}}
*{{cite arXiv |author-mask= 1 |first=Maxie D. |last=Schmidt |eprint=1702.01374 |title=Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers ''h'' ≥ 2|year=2017|class=math.CO }}
</ref>)
in several special cases of the prescribed sequences, {{math|''j<sub>n</sub>''}}, generated by the general expansions of the {{mvar|J}}-fractions defined in the first subsection. Here we define {{math|0 < {{abs|''a''}}, {{abs|''b''}}, {{abs|''q''}} < 1}} and the parameters <math>R, \alpha \isin \mathbb{Z}^+</math> and {{mvar|x}} to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these {{mvar|J}}-fractions are defined in terms of the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]], [[Pochhammer symbol]], and the [[binomial coefficients]].

{| class="wikitable"
|-
! <math>j_n</math> !! <math>c_1</math> !! <math>c_i (i \geq 2)</math> !! <math>\mathrm{ab}_i (i \geq 2)</math>
|-
| <math>q^{n^2}</math> || <math>q</math> || <math>q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)</math> || <math>q^{6h-10}\left(q^{2h-2}-1\right)</math>
|-
| <math>(a; q)_n</math> || <math>1-a</math> || <math>q^{h-1} - a q^{h-2} \left(q^{h} + q^{h-1} - 1\right)</math> || <math>a q^{2h-4} \left(a q^{h-2}-1\right)\left(q^{h-1}-1\right)</math>
|-
| <math>\left(z q^{-n}; q\right)_n</math> || <math>\frac{q-z}{q}</math> || <math>\frac{q^h - z - qz + q^h z}{q^{2h-1}}</math> || <math>\frac{\left(q^{h-1}-1\right) \left(q^{h-1}-z\right) \cdot z}{q^{4h-5}}</math>
|-
| <math>\frac{(a; q)_n}{(b; q)_n}</math> || <math>\frac{1-a}{1-b}</math> || <math>\frac{q^{i-2}\left(q+ab q^{2i-3}+a(1-q^{i-1}-q^i)+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}</math> || <math>\frac{q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^2\left(1-bq^{2i-3}\right)}</math>
|-
| <math>\alpha^n \cdot \left(\frac{R}{\alpha}\right)_n</math> || <math>R</math> || <math>R+2\alpha (i-1)</math> || <math>(i-1)\alpha\bigl(R+(i-2)\alpha\bigr)</math>
|-
| <math>(-1)^n \binom{x}{n}</math> || <math>-x</math> || <math>-\frac{(x+2(i-1)^2)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|-
| <math>(-1)^n \binom{x+n}{n}</math> || <math>-(x+1)</math> || <math>\frac{\bigl(x-2i(i-2)-1\bigr)}{(2i-1)(2i-3)}</math>
||<math>\begin{cases}-\dfrac{(x-i+2)(x+i-1)}{4 \cdot (2i-3)^2} & \text{for }i \geq 3; \\[4px] -\frac{1}{2}x(x+1) & \text{for }i = 2. \end{cases}</math>
|}

The radii of convergence of these series corresponding to the definition of the Jacobi-type {{mvar|J}}-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.