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Free martingale polynomials. (English) Zbl 1033.46050

Summary: We investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy 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.

MSC:

46L54 Free probability and free operator algebras
05A40 Umbral calculus
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
60G44 Martingales with continuous parameter
47H20 Semigroups of nonlinear operators
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