Fractional Pure birth processes

We consider a fractional version of the classical non-linear birth process of which the Yule-Furry model is a particular case. Fractionality is obtained by replacing the first-order time derivative in the difference-differential equations which govern the probability law of the process, with the Dzherbashyan-Caputo fractional derivative. We derive the probability distribution of the number $ \mathcal{N}_\nu \left( t \right)$ 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 relationship $\mathcal{N}_\nu \left( t \right) = \mathcal{N} \left( T_{2 \nu} \left( t \right) \right)$ where $\mathcal{N} \left( t \right)$ is the classical generalised birth process and $T_{2 \nu} \left( t \right)$ is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in section \ref{sec-exp} and various forms of its distribution are given and discussed.