Stieltjes, Poisson and other integral representations for functions of Lambert $W$

Understand

We show that many functions containing $W$ are Stieltjes functions.

• Explicit Stieltjes integrals are given for functions $1/W(z)$, $W(z)/z$, and others.
• We also prove a generalization of a conjecture of Jackson, Procacci & Sokal.
• Integral representations of $W$ and related functions are also given which are associated with the properties of their being Pick or Bernstein functions.