Using $P'_{n+1} - P'_{n-1} = (2n + 1)P_n$, show that $P'_{n+1}(x) = \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-2k) + 1)P_{n-2k}(x)$
$P_n(x)$ is legendre polynomial.
$$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x) + \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-1-2k)+1)P_{n-1-2k}(x))$$
How can I continue to use induction to prove this? Help appreciated.
https://math.indiana.edu/promotion/files/legendrepaperinprintCA.pdf proposition 2 from this paper could help