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A study on the bilinear equation of the sixth Painlev\'e transcendents
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The sixth Painlev\'e equation is a basic equation among the non-linear differential equations with three fixed singularities, corresponding to Gauss's hypergeometric differential equation among the linear differential equations. It is known that 2nd order Fuchsian differential equations with three singular points are reduced to the hypergeometric differential equations. Similarly, for nonlinear differential equations, we would like to determine the equation from the local behavior around the three singularities. In this paper, the sixth Painlev\'e equation is derived by imposing the condition that it is of type (H) at each three singular points for the homogeneous quadratic 4th-order differential equation.
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Refined Painlev\'e/gauge theory correspondence and quantum tau functions
Strong-coupling expansions of N=2 SU(2) partition functions in general Omega-background are derived from quantum Painlevé equations obtained via C2/Z2 blowup relations, and match refined holomorphic anomaly and irregu...
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