Title: Convergence Properties of Kemp’s q-Binomial Distribution

Author(s): Stefan Gerhold and Martin Zeiner
Issue: Volume 72 Series A Part 2 Year 2010
Pages: 331 -- 343
We consider Kemp's $q$-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are $q$-analogues of classical convergence properties. From the results about distributions, we deduce some new convergence results for ($q$-)Krawtchouk and $q$-Charlier polynomials. Besides elementary estimates, we apply Mellin transform asymptotics.
AMS (2000) subject classification. Primary 60F05; secondary 33D15.
Keywords and phrases: $q$-binomial distribution, discrete normal distribution, Heine distribution, $q$-Krawtchouk polynomials, $q$-Charlier polynomials, Mellin transform, limit theorems.