# Remarks on a free analogue of the beta prime distribution

## TL;DR

This paper introduces a free probability analogue of the classical beta prime distribution, constructed via free convolution, and explores its properties, moments, and relation to free Meixner distributions.

## Contribution

It defines the free beta prime distribution using free convolution and analyzes its moments and connection to free negative binomials, extending classical distribution concepts.

## Key findings

- Explicit calculation of moments using non-crossing linked partitions
- Identification of the free beta prime as part of the free Meixner family
- Development of free analogues of classical distributions like F, T, and beta distributions

## Abstract

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 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.