The free Meixner class for pairs of measures

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.