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Abstract:

The hypergeometric equation is a second-order ordinary differential equation with three regular singular points. It has been studied since the early nineteenth century, with important contributions from Leonhard Euler, Carl Friedrich Gauss, and Bernhard Riemann. This mini-course mainly follows the work of Bernhard Riemann and Ernst Kummer and their work with the transformation of the equation. The equation appears in several areas, including quantum physics, wave equations, and the theory of special functions.

The talk begins with an introduction to the hypergeometric equation and its solutions near each regular singular point, in the non-resonant case, which is in the work of Lazaruz Fuchs. These solutions are presented in both series and integral form, using the Gamma function. We then study how Möbius transformations act on the equation, and use the Riemann scheme to describe how both the equation and its solutions change under these transformations. The aim is to conclude the talk by explaining how a general Fuschian equation can be transformed into the hypergeometric equation through the Riemann scheme.