On the generalized iterates of Yeh's combinatorial K-species

AbstractLet f = f(x) = x + a2x2 + … ∈ K[[x]] be a “normalized” power series over a (commutative) field K of characteristic zero. The operator Δf: K[[x]] → K[[x]], defined by Δfg = g ∘ f − g, has been used in (G. Labelle, European J. Combin. 1 (1980), 113–138) to obtain formulas for the inverse f〈−1〉 and the generalized iterates f〈t〉, t ∈ K, of the series f. A. Joyal (in Lect. Notes in Math. Vol. 1234, pp. 126–159, Springer-Verlag, New York/Berlin, 1986) was the first to realize that Δf can be lifted to the combinatorial level. He made use of this fact to obtain a formula for a virtual species F〈−1〉 which is the inverse (under substitution) of any given normalized species F = X + …. Using the same operator, we show that the concept of K-species in the sense of Y.-N. Yeh (ibid.) (where K is now only a binomial half-ring) is a good context for the definition of the generalized iterates F〈t〉, t ∈ K, of any normalized species (or K-species). We present a new approach to Yeh's extension of substitution to K-species. We also introduce the notions of “infinitesimal generator,” “directional derivatives,” and “Lie bracket” of K-species, which turn out to be K-species, where K denotes the “rational closure” of K. These concepts give, in return, a better insight into substitution itself. For example, G ∘ F can be written in the form G ∘ F = (exp DΦ)G for a suitably chosen derivation DΦ. More generally, G ∘ F〈t〉 = (exp tDΦ)G. Two normalized K-species commute under substitution if and only if the Lie bracket of their infinitesimal generators is zero. Explicitly computed examples are also given