Let $P_n$ be the $n$th Bessel polynomial. Kelker (1971) showed that the Student $t$-distribution of $k = 2n + 1$ degrees of freedom is infinitely divisible if and only if $\varphi_n(x) = P_{n-1}(x^{\frac{1}{2}})/P_n(x^{\frac{1}{2}})$ is completely monotonic. Kelker and Ismail proved that $\varphi_n$ is indeed completely monotonic for some small values of $n$ and conjectured that this is always the case. This conjecture is proved here by a twofold application of Bernstein's theorem and the use of some special properties of the zeros of the Bessel polynomials. The same conclusion follows for $Y_k = (\chi k^2)^{-1}$, where $\chi k^2$ is a chi-square variable with $k$ degrees of freedom.
@article{1176996038,
author = {Grosswald, Emil},
title = {The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 680-683},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996038}
}
Grosswald, Emil. The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible. Ann. Probab., Tome 4 (1976) no. 6, pp. 680-683. http://gdmltest.u-ga.fr/item/1176996038/