Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 13E / b / Solution

A Papperitz symbol records the three regular singular points of a second-order Fuchsian equation and the two characteristic exponents at each point. Near $z=1$, substituting $w\sim (z-1{)}^{\rho }$ into the leading terms givesso the exponents are $0$ and $1/2$. The same calculation at $z=-1$ gives $0$ and $1/2$. At infinity, $w\sim z^s$ givesso $s=\pm n$. Since an exponent $\rho$ at infinity denotes behavior $z^{-\rho }$, these are recorded as $-n,n$. Thus the equation has P-symbolThe exponent sum is $1$, as required by the Fuchs relation for three singular points.