On Kendall–Ressel and related distributions

We delineate a connection of Kendall–Ressel and related laws with the lower real branch of Lambert WW function. A characterization of the canonical member of Kendall–Ressel class is found. The Letac–Mora interpretation of the reciprocity of two specific NEFs is extended by considering two related reproductive EDMs. A local limit theorem on gamma convergence for the reproductive back-shifted Kendall–Ressel EDM is derived. Each member of this EDM is self-decomposable and unimodal, but not strongly unimodal. The coefficient of variation, skewness and kurtosis of each representative of this EDM are higher than the corresponding measures for the members of gamma and inverse Gaussian EDMs. An integral representation for the lower real branch of Lambert WW function is given.