A class of Laplace transforms and their inversion (Q796780)

The author considers meromorphic functions of the form \(\Phi(s)=\Gamma(s)/\sum^{\infty}_{j=0}a_ j/(s+j)\) where \(a_ j (j\in N^ 0)\) are non-negative reals with \(0<\sum^{\infty}_{j=0}a_ j<\infty.\) It is shown that any such functions represents the bilateral Laplace transform \((\int^{\infty}_{-\infty}e^{-st} f(t) d\lambda(t))\) of a function \(f:R\to R.\) Furthermore, the inverse Laplace transform is determined by means of an inversion theorem for the bilateral transform. Finally two examples are given which are relevant to probability theory.