Letters in Mathematical Physics

, Volume 102, Issue 3, pp 297–321 | Cite as

Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus



We realize affine Weyl group symmetries on the Schrödinger equations for the quantum Painlevé equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schrödinger equations for the quantum Painlevé equations. In other words, since the Schrödinger equations for the quantum Painlevé equations are equivalent to the Knizhnik–Zamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik–Zamolodchikov equations.

Mathematics Subject Classification

17B80 33C70 34M55 81R12 81T40 


affine Weyl groups quantum Painlevé equations Knizhnik–Zamolodchikov equations hypergeometric integrals 


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Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Department of MathematicsKobe UniversityKobeJapan

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