Legendre polynomial - recurrence relations

[Joe20]

Homework Statement

As attached

Relevant Equations

As attached

Note: $P_n (x)$ 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.

 

[Office_Shredder]

To prove something by induction, you need to do two things:
Prove it's true for n=1.

Prove that if it's true for n=m, that implies it's true for n=m+1.

The first part at least is usually pretty easy. Why don't you try to get as far as you can and post your work where you get stuck?