TL;DR
This paper explores the relationship between middle convolutions of Fuchsian systems and integral transformations of Heun's equation, linking these concepts to the geometry of the sixth Painlevé equation.
Contribution
It establishes a connection between middle convolutions and integral transformations of Heun's equation within the context of Fuchsian systems and Painlevé equations.
Findings
Middle convolutions correspond to integral transformations of Heun's equation.
Heun's equation arises from Fuchsian systems with four singularities.
The study links the geometry of Painlevé equations to differential equations transformations.
Abstract
Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlev\'e equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.
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