Markov processes, polynomial martingales and orthogonal polynomials

We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales ${\{M_n(X_t,t)\}}_{n\ge 1}$ that contains complete information on the distribution (both marginal and transitional) of the process. We specify conditions expressed in terms of $M_n^{\mathrm{\prime }}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic harness. We also give conditions under which some of these martingales are also reversed martingales.