Stieltjes, Poisson and similar representations of functions

Abstract

We show that the principal branch of the Lambert TeX Embedding failed! function is a Bernstein function. We also show that many functions containing TeX Embedding failed!, for example TeX Embedding failed!, are Stieltjes functions. The fact that TeX Embedding failed! and related functions are Stieltjes, Pick or Bernstein functions leads naturally to integral representations for them, and various integral representations are obtained, including Poisson's integrals. Integral forms based on the Burniston--Siewert method are given as well. Finally, we prove a recent conjecture of B. Jackson, A. Procacci, and A.D. Sokal.

One of the integral representations has been known for some years now, and is on the Lambert~TeX Embedding failed! poster (http://www.orcca.on.ca/LambertW/), but to our knowledge the proof is heretofore unpublished. Several others are new to this work.

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