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

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

2006-07-28

arxiv.org/abs/nlin/0607065v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages

Scientific paper

A class of classical solutions to the $q$-Painlev\'e equation of type $(A_1+A_1')^{(1)}$ (a $q$-difference analog of the Painlev\'e II equation) is constructed in a determinantal form with basic hypergeometric function elements. The continuous limit of this $q$-Painlev\'e equation to the Painlev\'e 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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