The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation

We consider the fractional telegraph equation with partial fractional derivatives of rational order $\alpha=m/n$ with $m<n$. We prove that the fundamental solution to the Cauchy problem for this equation can be expressed as the distribution of the composition of two processes, one depending on $m$ (denoted by $T_{m}$) and the other one depending on $n$ (representing the "time"). In the special case where $m=1$, $T_{1}$ coincides with the classical telegraph process, while $T_{m}$, for $m>1$; is a telegraph process stopped at stable distributed times. We obtain explicit expressions for the probability distribution of a telegraph process with a random time and for the characteristic function of a telegraph process stopped at stable-distributed times.