Proving a property of Legendre polynomials containing its derivatives: $nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$

Question

I am trying to prove the following property of Legendre polynomials. $$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$ My guess is that I somehow have to use the Bonnets recursion formula $$(n+1)P_{n+1}(x)=(2n+1)xP_n(x)-nP_{n-1}(x)$$ which is proved using the generating function of the legendre polynomials. However, I am not being able to eliminate the derivative of $P_{n+1}$ from this formula. I am not being able to do problems of a similiar nature, of recursion relations between the derivative of legendre polynomials.

Answer 1

If $f(x,t)=\sum P_n(x)t^n=(1-2tx+t^2)^{-1/2}$ is the generating function then you want to show that $$t\frac{\partial}{\partial t}f=x\frac{\partial}{\partial x}f-t\frac{\partial}{\partial x}f$$ and it is true: $$(t\frac{\partial}{\partial t}+(t-x)\frac{\partial}{\partial x})(1-2tx+t^2)=0.$$