Conditional moments of <Emphasis Type="Italic">q</Emphasis>-Meixner processes

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes.Mathematics Subject Classification (2000): 60J25Research partially supported by NSF grant #INT-0332062, by the C.P. Taft Memorial Fund, and University of Cincinnatis Summer Faculty Research Fellowship ProgramAcknowledgementPart of the research of WB was conducted while visiting the Faculty of Mathematics and Information Science of Warsaw University of Technology. The authors thank M. Boejko for bringing to their attention several references, to Hiroaki Yoshida for information pertinent to Theorem 4.3, and to M. Anshelevich, W. Matysiak, R. Speicher, P. Szabowski, and M. Yor for helpful comments and discussions. Referees comments lead to several improvements in the paper.