Editing Generating function (section)

===Transformations of generating functions===
There are a number of transformations of generating functions that provide other applications (see the [[generating function transformation|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 [[Generating function transformation#Integral Transformations|integral transformations]]) or weighted sums over the higher-order derivatives of these functions (see [[Generating function transformation#Derivative Transformations|derivative transformations]]).

Generating function transformations can come into play when we seek to express a generating function for the sums
<math display="block">s_n := \sum_{m=0}^n \binom{n}{m} C_{n,m} a_m, </math>

in the form of {{math|''S''(''z'') {{=}} ''g''(''z'') ''A''(''f''(''z''))}} involving the original sequence generating function. For example, if the sums are
<math display="block">s_n := \sum_{k = 0}^\infty \binom{n+k}{m+2k} a_k \,</math>
then the generating function for the modified sum expressions is given by<ref>{{harvnb|Graham|Knuth|Patashnik|1994|p=535, exercise 5.71}}</ref>
<math display="block">S(z) = \frac{z^m}{(1-z)^{m+1}} A\left(\frac{z}{(1-z)^2}\right)</math>
(see also the [[binomial transform]] and the [[Stirling transform]]).

There are also integral formulas for converting between a sequence's OGF, {{math|''F''(''z'')}}, and its exponential generating function, or EGF, {{math|''F̂''(''z'')}}, and vice versa given by
<math display="block">\begin{align}
F(z) &= \int_0^\infty \hat{F}(tz) e^{-t} \, dt \,, \\[4px]
\hat{F}(z) &= \frac{1}{2\pi} \int_{-\pi}^\pi F\left(z e^{-i\vartheta}\right) e^{e^{i\vartheta}} \, d\vartheta \,,
\end{align}</math>

provided that these integrals converge for appropriate values of {{mvar|z}}.