Semigroups of distributions with linear Jacobi parameters

TL;DR

This paper characterizes when convolution semigroups of measures have polynomial Jacobi parameters, linking them to the Meixner class in classical, free, and two-state free convolutions, and explores their properties.

Contribution

It establishes the equivalence between polynomial Jacobi parameters and the Meixner class for classical, free, and two-state free convolutions, introducing new classes and properties.

Findings

Measures with polynomial Jacobi parameters are from the Meixner class.

Free convolution semigroups with polynomial Jacobi parameters are characterized explicitly.

Two-state free convolution semigroups with this property are constructed and shown to have Meixner-type features.

Abstract

We show that a convolution semigroup of measures has Jacobi parameters polynomial in the convolution parameter $t$ if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.