ABSTRACT. Species of structures were first introduced by Joyal as a unified framework for the theory of generating series in enumerative combinatorics. In 1988, Girard introduced normal functors as a model of pure lambda-calculus where terms are interpreted as infinite series with sets as coefficients (which correspond to a special case of Joyal's species). Fiore presented a generalized definition that both encompasses Joyal's species and constitutes a model of differential linear logic.

Since species encode families of labelled structures, much work has been done to investigate their connection with algebraic data types. We want to explore an alternative viewpoint of seeing them as terms in an extension of lambda-calculus motivated by the relationship between species and differential models of linear logic and use the combinatorial intuition as a guide in the design of the syntax. The next step would be to study methods of resolution of differential equations in the setting of generalized species with the ultimate goal being to establish in linear logic a combinatorial interpretation of the obtained differential calculus.