q-Probability: I. Basic Discrete Distributions

q-analogs of classical formulae go back to Euler, q-binomial coefficients were defined by Gauss, and q-hypergeometric series were found by E. Heine in 1846. The q-analysis was developed by F. Jackson at the beginning of the 20th century, and the modern point of view subsumes most of the old developments into the subjects of Quantum Groups and Combinational Enumeration. The general philosophy of q-analogs is that of a deformation, with the deformation parameter q being thought of as close to 1. This point of view is certainly not allencompassing; for example, representations of Quantum groups when q is a root of unity are of independent interests; more importantly, the q-pictures sometimes possess properties singular in (q − 1) or otherwise not regularly dependent on (q − 1); regularization of divergent/infinite (q = 1)−quantities is another useful feature of q-analogs... the list goes on. The typical example is