The free Meixner class for pairs of measures

TL;DR

This paper explores a new class of measure pairs with linear Jacobi parameters in two-state free convolution semigroups, revealing their Meixner-type properties and connections to limit theorems and quadratic harnesses.

Contribution

It introduces and analyzes a novel class of measures with Meixner-like properties in the context of two-state free convolution semigroups, extending previous work on classical and free Meixner classes.

Findings

The measures exhibit Meixner-type properties.

They appear in limit theorems related to convolution semigroups.

They are connected to the $q=0$ case of quadratic harnesses.

Abstract

We investigate in more detail the two-state free convolution semigroups of pairs of measures whose Jacobi parameters are linear in the convolution parameter $t$. These semigroups were constructed in arXiv:1001.1540, where we also showed that measures with the analogous property for the usual and free convolution are exactly the classical, resp. free Meixner classes. The class of measures in this paper has not been considered explicitly before, but we show that it also has Meixner-type properties. Specifically, it appears in limit theorems, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.