Integral representation of solutions to Fuchsian system and Heuns equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun’s differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard’s solution of the sixth Painlev'e equation, and to Heun’s equation.

💡 Research Summary

The paper develops explicit integral representations for solutions of certain Fuchsian systems and for Heun’s differential equation, and uses these representations to compute monodromy data in concrete cases. After a concise introduction that situates Fuchsian systems as first‑order linear ODEs with a finite set of regular singular points, the authors focus on the four‑point case (singularities at 0, 1, t, ∞) which is directly related to the second‑order Heun equation. By normalizing the trace of each residue matrix to zero (the Fuchs condition) the system is reduced to a 2 × 2 matrix form.

The core technical contribution is a Mellin–Barnes type contour integral formula for a fundamental matrix solution:

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