q-distributions and Markov processes

AbstractWe consider a sequence of integer-valued random variables Xn, n ⩾ 1, representing a special Markov process with transition probability λn, l, satisfying Pn, l = (1 − λn, l) Pn−1, l + λn, l−1 Pn−1, l−1. Whenever the transition probability is given by λn, l = qαn + βl + γ and λn, l = 1 − qαn+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 α, β and γ