q-distributions and Markov processes

We consider a sequence of integer{valued random variables Xn; n 1; representing a special Markov process with transition probability the transition probability is given by n;` = q n+`+ and n;` = 1 ? q n+`+ , we can nd closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q{binomial coeecients 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 ; and .