A generalization of the Kullback-Leibler divergence and its properties

A generalized Kullback-Leibler relative entropy is introduced starting with the symmetric Jackson derivative of the generalized overlap between two probability distributions. The generalization retains much of the structure possessed by the original formulation. We present the fundamental properties including positivity, metricity, concavity, bounds and stability. In addition, a connection to shift information and behavior under Liouville dynamics are discussed.