  We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and
$X_2(t)$ on the real line $\Bbb R$, both developing with finite constant speed
$c>0$, that, at the initial time instant $t=0$, simultaneously start from the
origin $0\in\Bbb R$ and whose evolutions are controlled by two independent
homogeneous Poisson processes of the same rate $\lambda>0$. Closed-form
expressions for the transition density $p(x,t)$ and the probability
distribution function $\Phi(x,t)=\text{Pr} \{ S(t)<x \}, \; x\in\Bbb R, \;
t>0,$ of the sum $S(t)=X_1(t)+X_2(t)$ of these processes at arbitrary time
instant $t>0$, are obtained. It is also proved that the shifted time derivative
$g(x,t)=(\partial/\partial t+2\lambda)p(x,t)$ satisfies the Goldstein-Kac
telegraph equation with doubled parameters $2c$ and $2\lambda$. From this fact
it follows that $p(x,t)$ solves a third-order hyperbolic partial differential
equation, but is not its fundamental solution. The general case is also
discussed.
