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Hypergeometric \(\tau \) functions of the \(q\)-Painlevé systems of type \((A_{2}+A_{1})^{(1)}\). (English) Zbl 1217.33027

Summary: We consider a \(q\)-Painlevé III equation and a \(q\)-Painlevé II equation arising from a birational representation of the affine Weyl group of type \((A_{2}+A_{1})^{(1)}\). We study their hypergeometric solutions on the level of \(\tau \) functions.

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33E17 Painlevé-type functions
39A13 Difference equations, scaling (\(q\)-differences)
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