Convolution of beta prime distribution

TL;DR

This paper derives new identities for the convolution of beta prime distributions, linking hypergeometric functions and probability, and applies these to establish properties like complete monotonicity and Turán's inequality.

Contribution

It introduces novel identities involving beta prime convolutions using hypergeometric series transformations and applies them to prove inequalities and monotonicity properties.

Findings

Identities for beta prime convolution involving hypergeometric functions

Proof of Turán's inequality for special functions

Monotonicity properties of hypergeometric function quotients

Abstract

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for ${}_3F_2(1)$. Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Tur\'an's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill's ratio is discussed in detail.