Editing Generating function (section)

=== 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>