The Bessel Polynomials and the Student t Distribution

The quotient \[ \frac{{{}_2 F_0 ( - n + 1,n; - ; - {1 / 2}\sqrt x )}}{{\sqrt {x_2 } F_0 ( - n,n + 1; - ; - {1 / 2}\sqrt x )}} \equiv \frac{{P_{n - 1} (\sqrt x )}}{{P_n (\sqrt x )}}\] arose in connection with the problem of the infinite divisibility of the Student t distribution. It is shown that ${{P_{n - 1} (\sqrt x )} / {P_n (\sqrt x )}}$ is completely monotonic in $[0,\infty )$ for $n = 4$, $5$ and $6$. This implies that the Student $t$ distribution is infinitely divisible for $9$, $11$ and $13$ degrees of freedom. We show that certain power sums of the zeros of the simple Bessel polynomials are zero. This is then used to show that for every $n = 0,1,2, \cdots $, there exists $\theta _n > 0$ such that the inverse Laplace transform of ${{P_{n - 1} (\sqrt x )} / {P_n (\sqrt x )}}$ is nonnegative in $[\theta _n ,\infty )$. This supports our conjecture that ${{P_{n - 1} (\sqrt x )} / {P_n (\sqrt x )}}$ is completely monotonic in $(0_n , \infty )$ for all $n$, and that the Student $t$ distribution is infinitely divisible for odd degrees of freedom.