A functional equation for the Mellin-Stieltjes transforms of gamma distributions (Q1079773)

For any probability measure p on \(B_+\) \((B_+:\) \(\sigma\)-algebra of the Borel subsets of the (strict) positive real axis \(R_+)\) the function \(\phi_ p\) given by \((1)\quad \phi_ p(\xi)=\int x^{\xi - 1}dP(x)\) is called the Mellin-Stieltjes transform of p. Two theorems are given which represent a characterization of Mellin-Stieltjes transforms \(\phi\) (\(\xi)\), \(\psi\) (\(\xi)\) of gamma distributions by a functional equation of the form \(h(\xi_ 1+\xi_ 2)=\sum^{v}_{j=0}\sum^{v}_{k=0}c_{jk}\phi_ j(\xi_ 1)\psi_ k(\xi_ 2)\) for their quotient functions \(\phi_ j(\xi)=\phi (\xi +j)/\phi (\xi),\) \(\psi_ j(\xi)=\psi(\xi +j)/\psi(\xi).\)