Riemann's differential equation

Remarks

Its singular points are $a$, $b$, $c$, all regular singular points, and $\infty$ is an ordinary point. The exponents are $\alpha, \alpha'$ at $a$, $\beta, \beta'$ at $b$ and $\gamma, \gamma'$ at $c$.

Check at $z = a$: multiplying by $(z - a)^2$ and letting $z \to a$, the indicial equation is $r(r - 1) + (1 - \alpha - \alpha') r + \alpha\alpha' = (r - \alpha)(r - \alpha') = 0$, so the exponents at $a$ are $\alpha$ and $\alpha'$.

Example: take the singular points $0$, $\infty$, $1$, using the limiting form in which the factors with $b$ are dropped as $b \to \infty$. With exponents $0, 1 - C$ at $0$, $A, B$ at $\infty$ and $0, C - A - B$ at $1$, the equation becomes the hypergeometric differential equation with parameters $A, B, C$.

Non-example: exponents with sum different from $1$ are not allowed, because the equation then acquires a fourth singular point at $\infty$. The condition on the sum is Fuchs's relation for three singular points.

The solutions are written with the Riemann P-symbol.