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A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, is the Lagrange inversion theorem for power series (even in the formal context).

Starting from the exact sequence

$0 \rightarrow \mathbb{C} \rightarrow \mathbb{C}((z)) \xrightarrow{D} \mathbb{C}((z)) \;\xrightarrow{ \mathrm{Res} }\; \mathbb{C} \rightarrow 0,$

and using the simple rules of the formal derivative D, and of the formal residue Res (that by definition is the linear form taking the formal Laurent series $f(z)=\sum_{k=m}^{\infty}f_k z^k \in \mathbb{C}((z))$ to the coefficient $f_{-1}$ of $z^{-1}$ ) one easily proves:

(Lagrange inversion formula): If $f(z)=\sum_{k=1}^{\infty}f_k z^k \in > \mathbb{C}[[z]]$ and $g(z)=\sum_{k=1}^{\infty}g_k z^k \in > \mathbb{C}[[z]]$ are composition inverse of each other, the coefficients of the (multiplicative) powers of $f$ and $g$ are linked by the formula

$n[z^n]g^k=k[z^{-k}]f^{-n}$,

and in particular (for k=1)

$[z^n]g=\frac{1}{n} \mathrm{Res}( > f^{-n} ).$

(to whom didn't know it: enjoy computing the power series expansion of the composition inverse of $f(z):=z+z^m$, or of $f(z):=z\exp(z),$ and of their powers).

My question: what are the generalization of this theorem in wider context. I mean, in the same way that, just to make one example, the archetypal ODE $u'=\lambda u$ procreates the theory of semigroups of evolution in Banach spaces.

Also, I'd be grateful to learn some new nice application of this classic theorem.

(notation: for $f=\sum_{k=m}^{\infty}f_k z^k \in \mathbb{C}((z))$ the symbol $[z^k]f$ stands, of course, for the coefficient $f_{k}$)

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See also mathoverflow.net/questions/32099/… –  Tom Copeland Nov 28 '11 at 22:43
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4 Answers

up vote 7 down vote accepted

The Lagrange inversion formula allows to give a combinatorial interpretation of the Jacobian conjecture. See for example the classic paper of Bass, Connell and Write, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, http://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15032-7/S0273-0979-1982-15032-7.pdf.

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This theorem has been proven by many people and in the context of many diverse fields of mathematics since Lagrange presented "Nouvelle methode pour resoudre les equations litterales par le moyen des series."

It is so ubiiquitous that I've often thought that getting all the proofs together in one place would capture in a unique and compelling way the ebb and flow of the styles, habits, aesthetics and interests of the mathematics community over time. Indeed, I started on this project and quickly discovered that it was more than I could possibly do on my own. Just starting the project however gave me an even deeper appreciation of works such as Dickson's History of the Theory of Numbers and Fletcher's Index of Mathematical Tables.

I've got a list of 50+ proofs, a large collection of books, papers and articles and a bibtex bibliography if anybody is intrigued.

IMHO, one of the most general statements is Krattenhaler's "Operator Methods and Lagrange Inversion: A Unified Approach to Lagrange Formulas." One of the most unique is Abdesselam's "A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula."

Cheers, Scott (sbg at acw dot com)

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Richard Stanley has written some excellent notes in EC2 ch. 5 p67 about the Lagrange inversion formula (Theorem 5.4.2 on p38).

Stanley provides reference to An introduction to the theory of infinite series by Thomas John I'Anson Bromwich (You can see the full 1908 text through google books and other editions here: http://books.google.com/books?q=editions:UOM39015064521290&id=ZY45AAAAMAAJ) where several applications are provided.

Stanley also provides reference for generalizations of the Lagrange inversion formula:

I.M. Gessel's Paper (combinatorial proof): http://portal.acm.org/citation.cfm?id=31572 Note: Gessel also gives a generalization of Lagrange inversion to noncommutative power series in A noncommutative generalization and q-analog of the Lagrange inversion formula.

See also D.W. Stanton's Survey: Recent results for the q-Lagrange inversion formula.

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There is a large combinatorial literature on this, with many extensions and applications (including, notably, the MacMahon Master theorem). I am giving below a few pointers so you can start - the real literature is very large.

1) multivariate version. See "Combinatorial Enumeration" by Goulden and Jackson. See also this Gessel's paper with a combinatorial proof (there are other comb. proofs), many references, etc.

2) $q$- and non-commutative extensions. There are several $q$-analogues due to Andrews (see here, Garsia (see here for ref and bijective proof), and Gessel (see here for both $q$- and non-commutative generalization). Finally, see this (more general) non-commutative extension using quasideterminants.

3) the version in the language of species (see also this MO question) - see this paper by Gessel and Labelle.

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