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Hypergeometric \(\tau \)-functions of the \(q\)-Painlevé system of type \(E_{7}^{(1)}\). (English) Zbl 1163.33321

Summary: We present the \(\tau \)-functions for the hypergeometric solutions to the \(q\)-Painlevé system of type \(E_{7}^{(1)}\) in a determinant formula whose entries are given by the basic hypergeometric function \(_{8}W_{7}\). By using the \(W(D_{5})\) symmetry of the function \(_{8}W_{7}\), we construct a set of twelve solutions and describe the action of \(^{~}W(D_{6}^{(1)})\) on the set.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33D60 Basic hypergeometric integrals and functions defined by them
33E17 Painlevé-type functions
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