  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 $k\geq 0$, 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 $k\leq n$ is obtained in terms of properties of the
representing measure of $f$.

Classical Analysis and ODEs