Editing Generating function (section)

== Tables of special generating functions ==
An initial listing of special mathematical series is found [[List of mathematical series|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.<ref>See also the ''1031 Generating Functions'' found in {{cite thesis |first=Simon |last=Plouffe |title=Approximations de séries génératrices et quelques conjectures |trans-title=Approximations of generating functions and a few conjectures |year=1992 |type=Masters |publisher=Université du Québec à Montréal |language=fr |arxiv=0911.4975}}</ref>

{{expand section|Lists of special and special sequence generating functions. The next table is a start|date=April 2017}}

{| class="wikitable"
|-
! Formal power series !! Generating-function formula !! Notes
|-
| <math>\sum_{n = 0}^\infty \binom{m+n}{n} \left(H_{n+m}-H_m\right) z^n</math> || <math>\frac{1}{(1-z)^{m+1}} \ln \frac{1}{1-z}</math> || <math>H_n</math> is a first-order [[harmonic number]]
|-
| <math>\sum_{n = 0}^\infty B_n \frac{z^n}{n!}</math> || <math>\frac{z}{e^z-1}</math> || <math>B_n</math> is a [[Bernoulli number]]
|-
| <math>\sum_{n = 0}^\infty F_{mn} z^n</math> || <math>\frac{F_m z}{1-(F_{m-1}+F_{m+1})z+(-1)^m z^2}</math> || <math>F_n</math> is a [[Fibonacci number]] and <math>m \in \mathbb{Z}^{+}</math>
|-
| <math>\sum_{n = 0}^\infty \left\{\begin{matrix} n \\ m \end{matrix} \right\} z^n</math> || <math>(z^{-1})^{\overline{-m}} = \frac{z^m}{(1-z)(1-2z)\cdots(1-mz)}</math> || <math>x^{\overline{n}}</math> denotes the [[rising factorial]], or [[Pochhammer symbol]] and some integer <math>m \geq 0</math>
|-
| <math>\sum_{n = 0}^\infty \left[\begin{matrix} n \\ m \end{matrix} \right] z^n</math> || <math>z^{\overline{m}} = z(z+1) \cdots (z+m-1)</math>
|-
| <math>\sum_{n = 1}^\infty \frac{(-1)^{n-1}4^n (4^n-2) B_{2n} z^{2n}}{(2n) \cdot (2n)!}</math> || <math>\ln \frac{\tan(z)}{z}</math>
|-
| <math>\sum_{n = 0}^\infty \frac{(1/2)^{\overline{n}} z^{2n}}{(2n+1) \cdot n!}</math> || <math>z^{-1} \arcsin(z)</math>
|-
| <math>\sum_{n = 0}^\infty H_n^{(s)} z^n</math> || <math>\frac{\operatorname{Li}_s(z)}{1-z}</math> || <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]] function and <math>H_n^{(s)}</math> is a generalized [[harmonic number]] for <math>\Re(s) > 1</math>
|-
| <math>\sum_{n = 0}^\infty n^m z^n</math> || <math>\sum_{0 \leq j \leq m} \left\{\begin{matrix} m \\ j \end{matrix} \right\} \frac{j! \cdot z^j}{(1-z)^{j+1}}</math> || <math>\left\{\begin{matrix} n \\ m \end{matrix} \right\}</math> is a [[Stirling number of the second kind]] and where the individual terms in the expansion satisfy <math>\frac{z^i}{(1-z)^{i+1}} = \sum_{k=0}^{i} \binom{i}{k} \frac{(-1)^{k-i}}{(1-z)^{k+1}}</math>
|-
| <math>\sum_{k < n} \binom{n-k}{k} \frac{n}{n-k} z^k</math> || <math>\left(\frac{1+\sqrt{1+4z}}{2}\right)^n + \left(\frac{1-\sqrt{1+4z}}{2}\right)^n</math> ||
|-
| <math>\sum_{n_1, \ldots, n_m \geq 0} \min(n_1, \ldots, n_m) z_1^{n_1} \cdots z_m^{n_m}</math> || <math>\frac{z_1 \cdots z_m}{(1-z_1) \cdots (1-z_m) (1-z_1 \cdots z_m)}</math> || The two-variable case is given by <math>M(w, z) := \sum_{m,n \geq 0} \min(m, n) w^m z^n = \frac{wz}{(1-w)(1-z)(1-wz)}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{s}{n} z^n</math> || <math>(1+z)^s</math> || <math>s \in \mathbb{C}</math>
|-
| <math>\sum_{n = 0}^\infty \binom{n}{k} z^n</math> || <math>\frac{z^k}{(1-z)^{k+1}}</math> || <math>k \in \mathbb{N}</math>
|-
|<math>\sum_{n = 1}^\infty \log{(n)} z^n</math>||<math>\left.-\frac{\partial}{\partial s}\operatorname{{Li}_s(z)}\right|_{s=0}</math>
|}