Hello Math Stack Exchange community,
I am currently studying Legendre Polynomials and I encountered a recurrence relation that I would like to derive from Rodrigues' Formula. The recurrence relation is:
$$ (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x) $$
where $P_n(x)$ represents the Legendre polynomial of degree $n$.
Rodrigues' formula for the Legendre polynomial $P_n(x)$ is given by:
$$ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2 - 1)^n] $$
I understand that establishing this recurrence relation directly from Rodrigues' formula involves several steps of differentiation and algebraic manipulation. I would appreciate any guidance or steps toward deriving this recurrence relation using Rodrigues' formula.
Thank you in advance for your assistance.