The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible

Abstract

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.