  In this paper we investigate the properties of the free Sheffer systems,
which are certain families of martingale polynomials with respect to the free
Levy processes. First, we classify such families that consist of orthogonal
polynomials; these are the free analogs of the Meixner systems. Next, we show
that the fluctuations around free convolution semigroups have as principal
directions the polynomials whose derivatives are martingale polynomials.
Finally, we indicate how Rota's finite operator calculus can be modified for
the free context.

Combinatorics;Classical Analysis and ODEs;Operator Algebras;Probability