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

TL;DR

This paper proves the existence of a unique positive solution for the alternative discrete Painlevé I equation and links it to special solutions of the second Painlevé equation involving the Airy function, providing new insights into their positivity properties.

Contribution

It establishes the uniqueness and positivity of solutions for the alt-dP_I equation and characterizes these solutions via special solutions of P_II involving the Airy function.

Findings

The alt-dP_I equation has a unique positive solution for all n ≥ 0.

This positive solution is expressed in terms of a special solution of P_II involving Ai(t).

Solutions involving only Ai(t) remain positive for all n ≥ 0 and t ≥ 0.

Abstract

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$_{\rm I}$.