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-1/2K_ν(x1/2)/Kν+1(x1/2) is completely monotonic for ν -1. Our approach proves the stronger and new result, that x-1/2K_ν (x1/2) /Kν+1(x1/2) is a completely monotonic function of x for all real ν. We also derive a new integral representation.
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Mourad E. H. Ismail (1977) studied this question.