Free martingale polynomials

Abstract

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.

AMS-LaTeX with Times fonts; 25 pages

References in corpus (2)

• $q$-Gaussian processes: non-commutative and classical aspects
• On the computation of spectra in free probability

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• Dual Lukacs regressions for non-commutative variables
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• Radial Bargmann representation for the Fock space of type B
• Semigroups of distributions with linear Jacobi parameters
• Characterizations of some free random variables by properties of conditional moments of third degree polynomials
• Random matrix model for free Meixner laws