Special Functions, Stieltjes Transforms and Infinite Divisibility
Mourad E. H. Ismail, Douglas H. Kelker · SIAM Journal on Mathematical Analysis · 1979
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 $x^{{{( u - \mu )} /2}} {{I_\mu (\sqrt x )} /{I_ u (\sqrt x )}}$ is a completely monotonic function of x when $\mu > u > - 1$. This result and the complete monotonicity of $x^{{{( u - \mu )} /2}} {{K_ u (\sqrt x )} /{K_\mu (\sqrt x )}}$, $\mu > u > - 1$, 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 ${{K_ u (b\sqrt x )} /{K_ u (a\sqrt x )}}$ and ${{I_ u (a\sqrt x )} /{I_ u (b\sqrt x )}}$ for $b > a > 0$ and $ u > - 1$. We also obtain some results on the zeros of combinations of modified Bessel functions.