Stieltjes and other integral representations for functions of Lambert W

We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci and Sokal. In addition, we consider the relationship of functions of W with the class of completely monotonic functions and show that W is a complete Bernstein function.