Appell polynomials and their relatives II. Boolean theory

Abstract

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.

Orthogonal polynomials with a resolvent-type generating function

“…It makes sense therefore to look at the polynomials with the generating function (1), which we call the free Sheffer polynomials, and in particular at the polynomials satisfying the conditions of Proposition 1, which we call the free Meixner polynomials. Here the adjective "free" refers to their relation to free probability [20]; see [3,4] for more details. These polynomials can also be described explicitly; see Theorem 4 of [3].…”

Weak universality of dynamical $$\Phi ^4_3$$ Φ 3 4 : non-Gaussian noise

“…. , k 3 ), we define an abstract monomial of degree |k| as 2) and the parabolic distance between z, z ∈ R 4 by…”

“…A characteristic property of Appell sequences is the following identity (see for instance [2], the equation between (14) and Remark 3 therein with X distributed according to the measure μ here and Y = 0)…”

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

“…Other definitions and notations for Appell sequences can be found in the existing literature (see, for example [2]). …”