  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 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 $\mathcal{N}_{\nu}(t)$ 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
$\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))$, where $\mathcal{N}(t)$ is the
classical generalized birth process and $T_{2\nu}(t)$ 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.
