Bessel Functions and the Infinite Divisibility of the Student $t$- Distribution

Using the representation theorem and inversion formula for Stieltjes transforms, we give a simple proof of the infinite divisibility of the student $t$-distribution for all degrees of freedom by showing that $x^{-\frac{1}{2}}K_\nu(x^{\frac{1}{2}})/K_{\nu+1}(x^{\frac{1}{2}})$ is completely monotonic for $\nu \geqq -1$. Our approach proves the stronger and new result, that $x^{-\frac{1}{2}}K_\nu (x^{\frac{1}{2}}) /K_{\nu+1}(x^{\frac{1}{2}})$ is a completely monotonic function of $x$ for all real $\nu$. We also derive a new integral representation.