nLab Lagrange inversion

Lagrange inversion denotes a formula for the power series coefficients of the compositional inverse of an analytic function $f$ around $0$ satisfying $f(0)=0$, or of a formal power series in one or several variables.

Wikipedia: Lagrange inversion theorem, formal power series

Related entries: Faà di Bruno formula, noncommutative symmetric function, Lambert W-function

• I.M. Gessel, A noncommutative generalization and a $q$-analog of the Lagrange inversion formula, Trans. Amer. Soc. 257 (1980), pp. 455–482, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A 45 (1987), pp. 178–196
• Ira Gessel, Gilbert Labelle, Lagrange inversion for species, J. Combin. Theory Ser. A 72 (1995), 95–117.

A homotopical algebra proof of the Lagrange inversion formula is exhibited in

• Vladimir Dotsenko, A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula, arxiv/1606.08222

There are relations to Faà di Bruno algebra:

• Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, Non-commutative Hopf algebra of formal diffeomorphisms, Adv. Math. 200:2 (2006) 479-524 pdf
• Jean-Paul Bultel, Combinatorial properties of the noncommutative Faà di Bruno algebra, J. of Algebraic Combinatorics 38:243–273 (2013) MR3081645

An approach to Lagrange inversion using Heisenberg-Weyl algebra is in

• O. V. Viskov, A random walk with a skip-free component and the Lagrange inversion formula, Theory Probab. Appl., 45(1), 164–172. (9 pages); O. V. Viskov, Obrasčenie stepennyh rjadov i formula Lagranža, Dokl. AN SSSR, 254:2 (1980) 269–271

category: analysis, combinatorics