Correlated systems and extensivity of entropy

We present a generalization of the binomial distribution associated with a sequence of positive numbers that can be related to imperfect detections and to nonlinear coherent states. It involves asymmetric and symmetric expressions of probabilities for win- loss sequences of trials and models highly correlated systems . Our approach is based on generating functions and presents constraints of non-negativeness. Poisson-like limits and the Leibniz triangle rules are considered and analyzed. Our generalizations are illustrated by various analytical and numerical examples. After studying the asymmetric and symmetric generalized binomial distributions we explore, analytically, their asymptotical behavior (with respect to the number of trials) and discuss the extensivity of some entropic forms for these correlated systems.