Faà di Bruno formula is a remarkable combinatorial formula for higher derivatives of a composition of functions. There are various modern approaches to the related mathematics, using Joyal’s theory of species, operads, graphs/trees, combinatorial Hopf algebras and so on.
We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.
In works of T. J. Robinson the formula is treated in the context of vertex algebras, calculus with formal power series and in logarithmic calculus, as well as in a connection to the umbral calculus:
Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, Non-commutative Hopf algebra of formal diffeomorphisms, Adv. Math. 200:2 (2006) 479-524 pdf
Kurusch Ebrahimi-Fard, Frederic Patras, Exponential renormalization, Annales Henri Poincare 11:943-971,2010, arxiv/1003.1679 doi
Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed.
Hector Figueroa, Jose M. Gracia-Bondia, Joseph C. Varilly, Faà di Bruno Hopf algebras, article at Springer eom, math.CO/0508337
Jean-Paul Bultel, Combinatorial properties of the noncommutative Faà di Bruno algebra, J. of Algebraic Combinatorics 38:243–273 (2013) MR3081645
We give a new combinatorial interpretation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder–Frabetti–Krattenthaler for the antipode of the noncommutative Faà di Bruno algebra.