On an infinitely divisible distribution involving modified Bessel functions

Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for $\nu >\mu \ge 0$ and $b>a>0$ the function $$x\mapsto G(x;\mu ,\nu )={\left(\frac{b}{a}\right)}^{\mu -\nu }\frac{I_{\mu }(a\sqrt{x})I_{\nu }(b\sqrt{x})}{I_{\mu }(b\sqrt{x})I_{\nu }(a\sqrt{x})},$$ where $I_{\nu }$ stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros $j_{\nu ,n}$ with respect to the order, where $j_{\nu ,n}$ denotes the $n$th positive zero of the Bessel function of the first kind of order $\nu$. In this paper our aim is to reduce the problem further to the monotonicity, with respect to $\nu$, of $$Q_{\nu }(t)=\sum _{n\ge 1}{j}_{\nu ,n}^2{e}^{-t{j}_{\nu ,n}^2},t>0,$$ and to prove this monotonicity. In this way we show that indeed the above function $x\mapsto G(x;\mu ,\nu )$ is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein’s theorem and a probability tail argument.