Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 7E / b

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.