Hypergeometric Solutions to the q-Painlevé Equation of Type $(A_1+A_1^{\prime }{)}^{(1)}$

Abstract

A class of classical solutions to the $q$-Painlevé equation of type $(A_1+A_1^{\prime }{)}^{(1)}$ (a $q$-difference analog of the Painlevé II equation) is constructed in a determinantal form with basic hypergeometric function elements. The continuous limit of this $q$-Painlevé equation to the Painlevé II equation and its hypergeometric solutions are discussed. The continuous limit of these hypergeometric solutions to the Airy function is obtained through a uniform asymptotic expansion of their integral representation.

17 pages