Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 14A / b

Let $\omega$ solve the equation represented by the original symbol and define a dependent-variable rescaling of a Papperitz symbolwith a consistent local choice of branch. Near $a$, the multiplier is a nonzero analytic factor times $(z-a{)}^{\delta }$, so both exponents at $a$ increase by $\delta$. Near $b$, it is a nonzero analytic factor times $(z-b{)}^{-\delta }$, so both exponents at $b$ decrease by $\delta$. It is analytic and nonzero near $c$, so the exponents there are unchanged. Therefore the transformed equation has symbolThe opposite shifts also preserve the Fuchs relation.