A q -Pólya urn model and the q -Pólya and inverse q -Pólya distributions

A q-Pólya urn model is introduced by assuming that the probability of drawing a white ball at a drawing varies geometrically, with rate q, both with the number of drawings and the number of white balls drawn in the previous drawings. Then, the probability mass functions and moments of (a) the number of white balls drawn in a specific number of drawings and (b) the number of black balls drawn until a specific number of white balls are drawn are derived. These two distributions turned out to be q-analogs of the Pólya and the inverse Pólya distributions, respectively. Also, the limiting distributions of the q-Pólya and the inverse q-Pólya distributions, as the number of balls in the urn tends to infinity, are shown to be a q-binomial and a negative q-binomial distribution, respectively. In addition, the positive or negative q-hypergeometric distribution is obtained as conditional distribution of a positive or negative q-binomial distribution, given its sum with another positive or negative q-binomial distribution, independent of it.