Semigroups of Distributions with Linear Jacobi Parameters Academic Article uri icon


  • We show that a convolution semigroup {μ t} 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. © 2012 Springer Science+Business Media, LLC.

author list (cited authors)

  • Anshelevich, M., & Młotkowski, W.

citation count

  • 4

publication date

  • February 2012