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The Bessel function ratios (b/a) ^ K_ (as^1{2}) (a > b > 0, R) and (b/a) ^ I_ (as^1{2}) /I_ (bs^1{2}) (0 -1) are infinitely divisible Laplace transforms in s > 0. These results are derived as hitting times of the Bessel diffusion process. The infinite divisibility of the t-distribution is deduced as a limiting result. A relationship with the von Mises-Fisher distribution is also demonstrated.
John T. Kent (Sun,) studied this question.