Stieltjes and other integral representations for functions of Lambert W

Abstract

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.

Keywords:

• Lambert W function
• Stieltjes functions
• completely monotonic functions
• Bernstein functions
• complete Bernstein functions

AMS Subject Classification::

• Primary: 33E99
• 30E20
• Secondary: 26A48

Acknowledgements

We are very grateful to Prof. Alan Sokal for his many interesting suggestions regarding the properties of the Lambert W function. We also thank him for many useful comments and constructive remarks, which greatly improved the original manuscript.