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

Taro Hamamoto·Kenji Kajiwara·Nicholas S. Witte
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Abstract

A class of classical solutions to the $q$-Painlevé equation of type $(A_1+A_1')^{(1)}$ (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.

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Taro Hamamoto, Kenji Kajiwara, Nicholas S. Witte. 2006-07-28. Hypergeometric Solutions to the q-Painlevé Equation of Type $(A_1+A_1')^{(1)}$. https://arxiv.org/abs/nlin/0607065

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