An infinitely divisible distribution involving modified Bessel functions

Abstract:

We prove that the function \[ {\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}\] is the Laplace transform of an infinitely divisible probability distribution when $v > \mu \geqslant 0$ and $b > a > 0$. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when $x$ lies in the right half-plane. We conjecture that \[ {\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}\] also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as $v \to \infty$, the infinite divisibility property holds for both functions.