### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 092, 18 pages      arXiv:1706.10087      https://doi.org/10.3842/SIGMA.2017.092

### A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$

Hidehito Nagao
Department of Arts and Science, National Institute of Technology, Akashi College, Hyogo 674-8501, Japan

Received July 03, 2017, in final form November 24, 2017; Published online December 10, 2017

Abstract
Recently a certain $q$-Painlevé type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlevé system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlevé type system and the $q$-Painlevé system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlevé type system are constructed in terms of the terminating $q$-hypergeometric function.

Key words: $q$-Painlevé system of type $E_7^{(1)}$; $q$-Garnier system; Padé method; $q$-hypergeometric function.

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