On a two-parameter Yule-Simon distribution

We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin [2019] in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in the mentioned paper. Finally, by superposing mutations to the branching process, we propose a model which leads to the full two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon [1955] on limiting word frequencies.