q-distributions and Markov processes

We consider a sequence of integer-valued random variables X_n, n 1, representing a special Markov process with transition probability λ_{n, l}, satisfying P_{n, l} = (1 ? λ_{n, l}) P_{n?1, l} + λ_{n, l?1} P_{n?1, l?1}. Whenever the transition probability is given by λ_{n, l} = q^{n + βl + γ} and λ_{n, l} = 1 ? q^n+βl+γ, we can find closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q-binomial coefficients and the q-Stirling numbers. In general, it turns out that the q-notation, up to now mainly used in the theory of q-hypergeometrical series, represents a powerful tool to deal with these kinds of problems. In this context we speak therefore about q-distributions. Finally, we present some possible, mainly graph theoretical interpretations of these random variables for special choices of , β and γ.