Free martingale polynomials

Details Free martingale polynomials Free martingale polynomials

2001-12-19

arxiv.org/abs/math/0112194v1

J. Funct. Anal. 201 (2003), 228-261

Mathematics

Combinatorics

AMS-LaTeX with Times fonts; 25 pages

Scientific paper

10.1016/S0022-1236(03)00061-2

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.

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