Editing Lagrange inversion theorem (section)

==Statement==

Suppose {{mvar|z}} is defined as a function of {{mvar|w}} by an equation of the form

:<math>z = f(w)</math>

where {{mvar|f}} is analytic at a point {{mvar|a}} and <math>f'(a)\neq 0.</math> Then it is possible to ''invert'' or ''solve'' the equation for {{mvar|w}}, expressing it in the form <math>w=g(z)</math> given by a [[power series]]<ref>{{cite book |editor=M. Abramowitz |editor2=I. A. Stegun |title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables |chapter=3.6.6. Lagrange's Expansion |place=New York |publisher=Dover |page=14 |year=1972 |url=http://people.math.sfu.ca/~cbm/aands/page_14.htm}}</ref>
:<math> g(z) = a + \sum_{n=1}^{\infty} g_n \frac{(z - f(a))^n}{n!}, </math>
where
:<math> g_n = \lim_{w \to a} \frac{d^{n-1}}{dw^{n-1}} \left[\left( \frac{w-a}{f(w) - f(a)} \right)^n \right]. </math>

The theorem further states that this series has a non-zero radius of convergence, i.e., <math>g(z)</math> represents an analytic function of {{mvar|z}} in a [[neighbourhood (mathematics)|neighbourhood]] of <math>z= f(a).</math> This is also called '''reversion of series'''.

If the assertions about analyticity are omitted, the formula is also valid for [[formal power series]] and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for {{math|''F''(''g''(''z''))}} for any analytic function {{mvar|F}}; and it can be generalized to the case <math>f'(a)=0,</math> where the inverse {{mvar|g}} is a multivalued function.

The theorem was proved by [[Joseph Louis Lagrange|Lagrange]]<ref>{{cite journal |author=Lagrange, Joseph-Louis |year=1770 |title=Nouvelle méthode pour résoudre les équations littérales par le moyen des séries |journal=Histoire de l'Académie Royale des Sciences et Belles-Lettres de Berlin |pages=251–326 |url=http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1768&seite:int=257}} https://archive.org/details/uvresdelagrange18natigoog/page/n13 (Note:  Although Lagrange submitted this article in 1768, it was not published until 1770.)</ref> and generalized by [[Hans Heinrich Bürmann]],<ref>Bürmann, Hans Heinrich, "Essai de calcul fonctionnaire aux constantes ad-libitum," submitted in 1796 to the Institut National de France. For a summary of this article, see: {{cite book |editor=Hindenburg, Carl Friedrich |title=Archiv der reinen und angewandten Mathematik |trans-title=Archive of pure and applied mathematics |location=Leipzig, Germany |publisher=Schäferischen Buchhandlung |year=1798 |volume=2 |chapter=Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann |trans-chapter=Attempt at a simplified analysis; an extract of an abridgement by Mr. Bürmann |pages=495–499 |chapter-url=https://books.google.com/books?id=jj4DAAAAQAAJ&pg=495}}</ref><ref>Bürmann, Hans Heinrich, "Formules du développement, de retour et d'integration," submitted to the Institut National de France. Bürmann's manuscript survives in the archives of the École Nationale des Ponts et Chaussées [National School of Bridges and Roads] in Paris. (See ms. 1715.)</ref><ref>A report on Bürmann's theorem by Joseph-Louis Lagrange and Adrien-Marie Legendre appears in:  [http://gallica.bnf.fr/ark:/12148/bpt6k3217h.image.f22.langFR.pagination "Rapport sur deux mémoires d'analyse du professeur Burmann,"] ''Mémoires de l'Institut National des Sciences et Arts: Sciences Mathématiques et Physiques'', vol. 2, pages 13–17 (1799).</ref> both in the late 18th century. There is a straightforward derivation using [[complex analysis]] and [[contour integration]];<ref>[[E. T. Whittaker]] and [[G. N. Watson]]. ''[[A Course of Modern Analysis]]''. Cambridge University Press; 4th edition (January 2, 1927), pp. 129–130</ref> the complex formal power series version is a consequence of knowing the formula for [[polynomial]]s, so the theory of [[analytic function]]s may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof, in that what is really needed is some property of the [[Formal power series#Formal residue|formal residue]], and a more direct formal [[Formal power series#The Lagrange inversion formula|proof]] is available. In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction.<ref>{{cite book | last1=Richard | first1=Stanley | 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><ref>{{Citation |last1=Ira|first1=Gessel |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>


If {{mvar|f}} is a formal power series, then the above formula does not give the coefficients of the compositional inverse series {{mvar|g}} directly in terms for the coefficients of the series {{mvar|f}}. If one can express the functions {{mvar|f}} and {{mvar|g}} in formal power series as

:<math>f(w) = \sum_{k=0}^\infty f_k \frac{w^k}{k!} \qquad \text{and}  \qquad  g(z) = \sum_{k=0}^\infty g_k \frac{z^k}{k!}</math>

with {{math|1=''f''<sub>0</sub> = 0}} and {{math|''f''<sub>1</sub> ≠ 0}}, then an explicit form of inverse coefficients can be given in term of [[Bell polynomial]]s:<ref>Eqn (11.43), p. 437, C.A. Charalambides, ''Enumerative Combinatorics,'' Chapman & Hall / CRC, 2002</ref>

:<math> g_n = \frac{1}{f_1^n} \sum_{k=1}^{n-1} (-1)^k n^\overline{k} B_{n-1,k}(\hat{f}_1,\hat{f}_2,\ldots,\hat{f}_{n-k}), \quad n \geq 2, </math>

where
:<math>\begin{align}
  \hat{f}_k &= \frac{f_{k+1}}{(k+1)f_{1}}, \\
   g_1 &= \frac{1}{f_{1}}, \text{ and} \\
   n^{\overline{k}} &= n(n+1)\cdots (n+k-1)
 \end{align}</math>
is the [[rising factorial]]. 

When {{math|1=''f''<sub>1</sub> = 1}}, the last formula can be interpreted in terms of the faces of [[Associahedron|associahedra]]  <ref>{{cite arXiv|eprint=1709.07504|class=math.CO|title=Hopf monoids and generalized permutahedra|last1=Aguiar|first1=Marcelo|last2=Ardila|first2=Federico|year=2017}}</ref>

:<math> g_n = \sum_{F \text{ face of } K_n} (-1)^{n-\dim F} f_F , \quad n \geq 2, </math> 

where <math> f_{F} = f_{i_{1}} \cdots f_{i_{m}} </math> for each face <math> F = K_{i_1} \times \cdots \times K_{i_m} </math> of the associahedron <math> K_n .</math>