The Bessel Polynomials and the StudenttDistribution

The quotient \[ {{{}_2 F_0 ( - n + 1,n; - ; - {1 / 2}√ x )}}{{√x_2 F_0 ( - n,n + 1; - ; - {1 / 2}√ x )}} ≡ {{Pn - 1 (√ x )}}{{P_n (√ x )}}\] arose in connection with the problem of the infinite divisibility of the Student t distribution. It is shown that Pn - 1 (√ x ) / Pₙ (√ x ) is completely monotonic in [0,∞ ) 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, ⋯, there exists θ ₙ > 0 such that the inverse Laplace transform of Pn - 1 (√ x ) / Pₙ (√ x ) is nonnegative in [θ ₙ ,∞ ). This supports our conjecture that Pn - 1 (√ x ) / Pₙ (√ x ) is completely monotonic in (0ₙ , ∞ ) for all n, and that the Student t distribution is infinitely divisible for odd degrees of freedom.