A Fuchsian (differential) equation is a linear homogeneous ordinary differential equation with analytic coefficients in the complex domain whose singular points are all regular singular points. In other words, it is the 1-dimensional case of the theory of meromorphic differential equations with only regular singular points. Hilbert's 21st problem is concerned with finding a Fuchsian equation with prescribed points of singularities and prescribed monodromies. The corresponding connection is also called Fuchsian.
M.V. Fedoryuk, Fuchsian equation, Springer online Enc. of Math.
J.L. Fuchs, J. Reine Angew. Math. 66, 121–160 (1866); 68, 354–385 (1868)
A. Hefliger, Local theory of meromorphic connections in dimension one (Fuchs theory), pages 129-149 of “Algebraic D-modules”, A. Borel, ed.
P. Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)
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