The primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input-output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators (Fliess operators) and analytic functions provided that the composite system is well defined and resides in one of these two classes. In each case, the radius of convergence of the interconnection is computed.