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, 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.
Doron Zeilberger, Toward a combinatorial proof of the Jacobian conjecture? in Combinatoire énumérative_ (Montreal, Que., 1985/Quebec, Que., 1985), 370–380, Lecture Notes in Math. 1234, Springer 1986. MR89c:05009
Eliahu Levy, Why do partitions occur in Faa di Bruno’s chain rule for higher derivatives?, math.GM/0602183.
E. Di Nardo, G. Guarino, D. Senato, A new algorithm for computing the multivariate Faà di Bruno’s formula, arxiv/1012.6008
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:
Thomas J. Robinson, New perspectives on exponentiated derivations, the formal Taylor theorem, and Faà di Bruno’s formula, Proc.Conf.Vert.Op.Alg., Cont.Math. 497 (2009) 185-198 arxiv/0903.3391; Formal calculus and umbral calculus, Electronic Journal of Combinatorics, 17(1) (2010) R95 arxiv/0912.0961
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.
Revised on March 16, 2013 15:11:15
by Tim Porter