Unique positive solution for an alternative discrete Painlev\'e I equation

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