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$_{ \rm I}$.
Categories: nlin.SI math.CA