Remarks on a Free Analogue of the Beta Prime Distribution

  • Hiroaki YoshidaEmail author


We introduce the free analogue of the classical beta prime distribution by the multiplicative free convolution of the free Poisson and the reciprocal of free Poisson distributions, and related free analogues of the classical F, T, and beta distributions. We show the rationales of our free analogues via the score functions and the potentials. We calculate the moments of the free beta prime distribution explicitly in combinatorial fashion by using non-crossing linked partitions and demonstrate that the free beta prime distribution belongs to the class of the free negative binomials in the free Meixner family.


Free beta prime distribution Free F-distribution Free T-distribution Free beta distribution Non-crossing linked partitions Free Meixner family 

Mathematics Subject Classification (2010)

46L54 60E05 



The author is very grateful to an anonymous referee who has read carefully and checked formulae and pointed out inaccurate parts in the original submission. The author also thanks T. Hasebe and N. Sakuma for their helpful discussions. This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) JP26400112.


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Authors and Affiliations

  1. 1.Department of Information SciencesOchanomizu UniversityBunkyoJapan

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