Fractional pure birth processes

We consider a fractional version of the classical nonlinear birth process of which the Yule Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the differencedifferential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N(v)(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N(v)(t) = N(T(2v) (t)), where N(t) is the classical generalized birth process and T(2v) (t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.