Unique positive solution for an alternative discrete Painlevé I equation

Abstract

We show that the alternative discrete Painlevé I equation (alt-dP${}_{\mathrm{I}}$) has a unique solution which remains positive for all $n\ge 0$. Furthermore, we identify this positive solution in terms of a special solution of the second Painlevé equation (P${}_{\mathrm{II}}$) involving the Airy function $\mathrm{Ai}(t)$. The special-function solutions of P${}_{\mathrm{II}}$ involving only the Airy function $\mathrm{Ai}(t)$ 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${}_{\mathrm{II}}$ and alt-dP${}_{\mathrm{I}}$.

21 pages, 6 figures

References in corpus (1)

• A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions

Cited by in corpus (8)

• On Airy Solutions of the Second Painlevé Equation
• Generalised Airy Polynomials
• Large $z$ Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane
• Unique positive solutions to $q$-discrete equations associated with orthogonal polynomials
• Nonlinear $q$-Stokes phenomena for $q$-Painlevé I
• Unique special solution for discrete Painlevé II
• Special solutions of a discrete Painlevé equation for quantum minimal surfaces
• Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV