Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su(2) coherent states, we propose interpretation of su(1, 1) and su(r, 1) coherent states in terms of probability theory. They will be called the negative binomial (mul...

We introduce a family of q-analogues of the binomial distribution, which generalises the Stieltjes-Wigert-, Rogers-Szegö-, and Kemp-distribution. Basic properties of this familiy are provided and several convergence results involving the classical binomial, Poisson, discrete normal distribution, and a family of q-analogues of the Poisson distribution are established. These results generalise co...

Abstract: This paper introduces three generalized geometric distributions: the binomial, negative binomial and Poisson distribution of the same order k. The generating functions and probability distributions of them are investigated, and then the corresponding modes of the distributions are discussed. By the Fibonacci sequence, the modes of the negative binomial distribution of order k are deri...