# Unique positive solution for an alternative discrete Painlevé I equation

Clarkson, Peter, Loureir0, Ana F., Van Assche, Walter (2016) Unique positive solution for an alternative discrete Painlevé I equation. Journal of Difference Equations and Applications, 22 (5). pp. 656-675. ISSN 1023-6198. E-ISSN 1563-5120. (doi:10.1080/10236198.2015.1127917) (KAR id:51098)

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## Abstract

We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second Painlevé equation and the alternative discrete Painlevé I equation.

Item Type: Article 10.1080/10236198.2015.1127917 Discrete Painleve equation, Painlevé equation, Airy function, Orthogonal polynomials Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA351 Special functionsQ Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics Peter Clarkson 20 Oct 2015 10:08 UTC 12 Feb 2020 04:07 UTC https://kar.kent.ac.uk/id/eprint/51098 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-8777-5284 https://orcid.org/0000-0002-4137-8822