Editing Formal power series (section)

===The Lagrange inversion formula===
{{main|Lagrange inversion theorem}}
As mentioned above, any formal series <math>f \in K[[X]]</math> with ''f''<sub>0</sub> = 0 and ''f''<sub>1</sub> ≠ 0 has a composition inverse <math>g \in K[[X]].</math> The following relation between the coefficients of ''g<sup>n</sup>'' and ''f''<sup>−''k''</sup> holds ("{{Visible anchor|Lagrange inversion formula}}"):

:<math>k[X^k] g^n=n[X^{-n}]f^{-k}.</math>

In particular, for ''n''&nbsp;=&nbsp;1 and all ''k''&nbsp;≥&nbsp;1,

:<math>[X^k] g=\frac{1}{k} \operatorname{Res}\left( f^{-k}\right).</math>

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting one proof here.{{efn|A number of different proofs exist, using techniques including Cauchy's coefficient formula for holomorphic functions, tree-counting arguments, or induction.<ref>{{cite book | last1=Stanley | first1=Richard | title=Enumerative combinatorics. Volume 1. | series =Cambridge Stud. Adv. Math. | volume=49 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-1-107-60262-5 | mr=2868112 }}</ref>{{pages?|date=March 2025}}<ref>{{Citation |last1=Gessel |first1=Ira|date=2016 |title=Lagrange inversion |journal=Journal of Combinatorial Theory, Series A |volume=144 |language=en |pages=212–249 |doi=10.1016/j.jcta.2016.06.018 |arxiv=1609.05988|mr=3534068}}</ref><ref>{{Citation |last1=Surya|first1=Erlang |last2=Warnke |first2=Lutz |date=2023 |title=Lagrange Inversion Formula by Induction |journal=The American Mathematical Monthly |volume=130 |issue=10 |language=en |pages=944–948 |doi=10.1080/00029890.2023.2251344 |arxiv=2305.17576|mr=4669236}}</ref>}} Noting <math>\operatorname{ord}(f) =1 </math>, we can apply the rules of calculus above, crucially Rule (iv) substituting <math>X \rightsquigarrow f(X)</math>, to get:

:<math>
\begin{align}
k[X^k] g^n & 
\ \stackrel{\mathrm{(v)}}=\ 
k\operatorname{Res}\left( g^n X^{-k-1}  \right)
\ \stackrel{\mathrm{(iv)}}=\ 
k\operatorname{Res}\left(X^n f^{-k-1}f'\right)
\ \stackrel{\mathrm{chain}}=\ 
-\operatorname{Res}\left(X^n (f^{-k})'\right) \\
& 
\ \stackrel{\mathrm{(ii)}}=\ 
\operatorname{Res}\left(\left(X^n\right)' f^{-k}\right)
\ \stackrel{\mathrm{chain}}=\ 
n\operatorname{Res}\left(X^{n-1}f^{-k}\right)
\ \stackrel{\mathrm{(v)}}=\ 
n[X^{-n}]f^{-k}.
\end{align}
</math>

'''Generalizations.''' One may observe that the above computation can be repeated plainly in more general settings than ''K''((''X'')): a generalization of the Lagrange inversion formula is already available working in the <math>\Complex((X))</math>-modules <math>X^{\alpha}\Complex((X)),</math> where α is a complex exponent. As a consequence, if ''f'' and ''g'' are as above, with <math>f_1=g_1=1</math>, we can relate the complex powers of ''f'' / ''X'' and ''g'' / ''X'': precisely, if α and β are non-zero complex numbers with negative integer sum, <math>m=-\alpha-\beta\in\N,</math> then

:<math>\frac{1}{\alpha}[X^m]\left( \frac{f}{X} \right)^\alpha=-\frac{1}{\beta}[X^m]\left( \frac{g}{X} \right)^\beta.</math>

For instance, this way one finds the power series for [[Lambert W function#Integer and complex powers|complex powers of the Lambert function]].