%0 Journal Article
%T Unique positive solution for an alternative discrete Painlev¨¦ I equation
%A Peter A. Clarkson
%A Ana F. Loureiro
%A Walter Van Assche
%J Physics
%D 2015
%I arXiv
%X 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$_{\rm I}$.
%U http://arxiv.org/abs/1508.04916v2