Editing Generating function (section)

==== 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]].