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Appell polynomials and their relatives. II: Boolean theory. (English) Zbl 1176.46060

Summary: The Appell-type polynomial family corresponding to the simplest noncommutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal noncommutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi–Nica and Bercovici–Pata maps, conditional freeness, and the Laha–Lukacs type characterization. A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi parameters under convolution, the relationship between the Jacobi parameters and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.
[For Part I, see Int.Math.Res.Not.2004, No.65, 3469–3531 (2004; Zbl 1086.33012).]

MSC:

46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
30B70 Continued fractions; complex-analytic aspects

Citations:

Zbl 1086.33012
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