Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 7E (Further Complex Methods)

A Riemann P-symbol specifies a second-order Fuchsian equation with regular singular points $a,b,c$ and the two local exponents shown beneath each point. The Fuchs relation is $\alpha +\alpha \prime +\beta +\beta \prime +\gamma +\gamma \prime =1$.

Set $z\prime =(1-z)/2$, sending $1,-1,\infty$ to $0,1,\infty$. Then rescale the dependent variable byThis shifts the exponent pairs to $(0,m)$ at zero and $(0,-m)$ at one, while leaving the exponents at infinity unchanged.

The hypergeometric exponent differences give $1-c=m$, $c-a-b=-m$, and exponents $a,b$ at infinity. Hence one may takeso a solution is ${}_2{F}_1(n,1-n;1-m;(1-z)/2)$, with the prefactor from part (i) when returning to the original dependent variable.