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.

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

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 $z^{-\rho }$.