Past exam of the analysis course of the University of Oxford / 2009 / B / 4 / iii / Solution

Expand both generating functions and use orthogonality of the Legendre polynomials, with ${\int }_{-1}^1{P}_n{P}_m=2{\delta }_{nm}/(2n+1)$. The allowed exchanges giveHence it depends only on $r=wz$. Explicitly,The series form alone already proves the required dependence.