  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.
