Special Functions, Stieltjes Transforms and Infinite Divisibility

Abstract

We establish the complete monotonicity of several quotients of Whittaker (Tricomi) functions and of parabolic cylinder functions. These results are used to show that the F distribution of any positive degrees of freedom (including fractional) is infinitely divisible and self-decomposable. We also prove the infinite divisibility of several related distributions, including the square of a gamma variable. We also prove that $x^{{{(\nu - \mu )} /2}} {{I_\mu (\sqrt x )} /{I_\nu (\sqrt x )}}$ is a completely monotonic function of x when $\mu > \nu > - 1$. This result and the complete monotonicity of $x^{{{(\nu - \mu )} /2}} {{K_\nu (\sqrt x )} /{K_\mu (\sqrt x )}}$, $\mu > \nu > - 1$, are used to introduce two new continuous infinitely divisible probability distributions. The limiting cases contain the reciprocal of a gamma distribution and a distribution whose probability density function is a “generalized” theta function. The first distribution is used as a mixing distribution to introduce a new, two parameter, symmetric, infinitely divisible probability distribution on the real line, which contains the Student t distribution as a limiting case. We also establish the complete monotonicity of ${{K_\nu (b\sqrt x )} /{K_\nu (a\sqrt x )}}$ and ${{I_\nu (a\sqrt x )} /{I_\nu (b\sqrt x )}}$ for $b > a > 0$ and $\nu > - 1$. We also obtain some results on the zeros of combinations of modified Bessel functions.