Appell polynomials and their relatives II. Boolean theory

TL;DR

This paper explores Boolean Appell polynomials, revealing their properties, generating functions, and connections to Meixner classes, with applications to non-commutative probability maps and cumulant analysis.

Contribution

It establishes the properties of Boolean Appell polynomials, showing their relation to free and classical Meixner classes and extending results to multivariate cases.

Findings

Boolean Appell polynomials have resolvent-type generating functions.

Boolean and free Meixner classes coincide even multivariately.

Properties like Jacobi coefficients behavior extend from free and classical to Boolean theory.

Abstract

The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative 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 coefficients under convolution, the relationship between the Jacobi coefficients 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.