Papperitz symbol

A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.

This symbol records the local exponents of Riemann's differential equation.

For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.

For an order-$n$ Fuchsian differential equation with $r$ finite singular points and infinity, the sum of all characteristic exponents is $n(n-1)(r-1)/2$. The three-singularity second-order case has $n=2$ and $r=2$.

A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.

Multiplying a solution by $(z-z_0{)}^{\lambda }$ adds $\lambda$ to both exponents at the finite point $z_0$ and subtracts $\lambda$ from both exponents at infinity, under the convention that an exponent $\rho$ at infinity corresponds to behavior $z^{-\rho }$.