On Generalized Stieltjes Functions

Abstract

It is shown that a function f is a generalized Stieltjes function of order $$\lambda >0$$λ>0if and only if $$x^{1-\lambda }(x^{\lambda -1+k}f(x))^{(k)}$$x1-λ(xλ-1+kf(x))(k)is completely monotonic for all $$k\ge 0$$k≥0, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions f for which $$x^{1-\lambda }(x^{\lambda -1+k}f(x))^{(k)}$$x1-λ(xλ-1+kf(x))(k)is completely monotonic for all $$k\le n$$k≤nis obtained in terms of properties of the representing measure of f.