  We show that the alternative discrete Painlev\'e I equation (alt-dP$_{\rm
I}$) has a unique solution which remains positive for all $n \geq 0$.
Furthermore, we identify this positive solution in terms of a special solution
of the second Painlev\'e equation (P$_{\rm II}$) involving the Airy function
$\mathop{\rm Ai}(t)$. The special-function solutions of P$_{\rm II}$ involving
only the Airy function $\mathop{\rm Ai}(t)$ therefore have the property that
they remain positive for all $n\geq 0$ and all $t \geq 0$, which is a new
characterization of these special solutions of P$_{\rm II}$ and alt-dP$_{\rm
I}$.
