Generalized binomial coefficients for molecular species

Let xi be a complex variable. We associate a polynomial in xi, denoted((M)( N))(xi), to any two molecular species M = M(X) and N = N(X) by means of a b inomial-type expansion of the form M(xi+X) = Sigma(N)((M)(N))(xi) N(X) In the special case M(X) = X-m, the species of linear orders of length m, t he above formula reduces to the classical binomial expansion (xi+X)(m) = Sigma(n)((m)(n)) xi(m-n)X(n). When delta = 1, a M(1 + X)-structure can be interpreted as a partially labe lled M-structure and ((M)(N))(1) is a nonnegative integer, denoted ((M)(N)) for simplicity. We develop some basic properties of these "generalized bin omial coefficients" and apply them to study solutions, Phi, of combinatoria l equations of the form M(Phi) = Psi in the context of C-species, M being m olecular and Psi being a given C-species. This generalizes the study of sym metric square roots (where M = E-2, the species of 2-element sets) initiate d by P. Bouchard, Y. Chiricota, and G. Labelle in (1995, Discrete Math. 139 , 49-56). (C) 2000 Academic Press.