Appell polynomials and their relatives II. 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.

References in corpus (7)

• On a class of free Levy laws related to a regression problem
• Quadratic bosonic and free white noises
• Appell polynomials and their relatives
• Orthogonal polynomials with a resolvent-type generating function
• Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples
• On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
• Eta-series and a Boolean Bercovici-Pata bijection for bounded k-tuples

Cited by in corpus (11)

• An infinite dimensional umbral calculus
• Convolution, subordination and characterization problems in noncommutative probability
• On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
• A new proof for the multiplicative property of the boolean cumulants with applications to operator-valued case
• Semigroups of distributions with linear Jacobi parameters
• Generators of some non-commutative stochastic processes
• A quadratic regression problem for two-state algebras with application to the Central Limit Theorem
• New characterization of two-state normal distribution
• On the asymptotics of the rescaled Appell polynomials
• Appell polynomials and their relatives III. Conditionally free theory
• Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability