Geodesic
Fuchsian Equations with Three Non-Apparent Singularities
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.
Keywords: Fuchsian equations; hypergeometric equation; difference equations; apparent singularities; bispectral duality; positive curvature; conic singularities. 
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Alexandre Eremenko; Vitaly Tarasov. Fuchsian Equations with Three Non-Apparent Singularities. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a57/

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