Riemann's differential equation [SEP] named after
Riemann's differential equation
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points (RSPs) to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and formula_1. The equation is also known as the Papperitz equation. 
The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and formula_1. That equation admits two linearly independent solutions; near a
Similarly, the exponents for are and for the two solutions.
One has thus
where the condition (sometimes called Riemann's identity)
on the exponents of the solutions of Riemann's differential equation has been used to define .
The first set of constants on the left hand side in , denotes the regular singular points of Riemann's differential equation. The second set, , are the corresponding exponents at for one of the two linearly independent solutions, and, accordingly, are exponents at for the second solution.