Lambert's \(W\), infinite divisibility and Poisson mixtures (Q633650)

The author considers Lambert's \(W\)-function, defined as the solution of the equation \[ We^{W} = x. \] It is shown that \(W\) is the Laplace exponent of an infinitely divisible distribution on \((0, \infty)\). The corresponding distribution is called Lambert law. It is shown to be an exponential mixture, and its cumulants are determined explicitly. Three families of Poisson mixture laws are shown to be based on Lamberts law, and several connections with known distributions are established. There is some overlap with a paper by \textit{L. Bondesson} and the reviewer [J. Math. Anal. Appl. 295, No. 1, 134--143 (2004; Zbl 1054.60018)].