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 binomial-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, the 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 labelled M-structure and ((M)(N))(1) is a nonnegative integer, denoted ((M)(N)) for simplicity. We develop some basic properties of these "generalized binomial coefficients" and apply them to study solutions, Phi, of combinatorial equations of the form M(Phi) = Psi in the context of C-species, M being molecular and Psi being a given C-species. This generalizes the study of symmetric square roots (where M = E-2, the species of 2-element sets) initiated by P. Bouchard, Y. Chiricota, and G. Labelle in (1995, Discrete Math. 139, 49-56). (C) 2000 Academic Press. [References: 15]
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