On generalized Stieltjes functions

Abstract

It is shown that a function $f$ is a generalized Stieltjes function of order $\lambda >0$ if and only if $x^{1-\lambda }(x^{\lambda -1+k}f(x){)}^{(k)}$ is completely monotonic for all $k\ge 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-\lambda }(x^{\lambda -1+k}f(x){)}^{(k)}$ is completely monotonic for all $k\le n$ is obtained in terms of properties of the representing measure of $f$.

17 pages