# Using Frobenius method to solve the Legendre differential equation

I'm tasked with solving the Legendre differential equation, and

Using $$c=0$$, obtain a series of even powers of $$x$$ (with $$a_1=0$$).

I found this exercise to be good at highlighting what I found confusing about this method. Firstly, I find the following indicial equations:

$$a_0 (c-1)c = 0$$ $$a_1(c+1)c = 0$$

I am instructed to choose $$c=0$$. Doing so, I don't see how this implies $$a_1=0.$$ Setting $$c=0$$ instantly satisfies the second equation $$\forall a_1 \in \mathbb R$$. If I chose $$c=1$$, that would indeed imply $$a_1 = 0$$, but I don't see how $$c=0$$ does. That aside, I end up with the canonical recurrence relation

$$a_{n+2} = \frac{n(n+1) - l(l+1)}{(n+2)(n+1)}a_n$$

And I went to check that even and odd indices of $$a_n$$ were non-zero. They were, so I don't see how I need to find an odd and even part for $$c=0$$. By forcing $$a_1=0$$, I can express an even part just by observing the even terms of recurrence relation. However, further I am tasked with:

Using $$c=1$$, obtain a series of odd powers of $$x$$ (with $$a_1=0$$).

Doing this forces $$a_1$$ to be zero, and I would then think that I'd only have odd powers of $$x$$ since $$a_1 = 0 \implies a_{n + 2} = 0 \implies a_3 = a_5 = ... = 0$$. However, I am meant to prove the recurrence relation:

$$a_{n+2} = \frac{(n+1)(n+2) - l(l+1)}{(n+2)(n+3)}a_n$$

Which I don't recognize. This here highlights my confusions with this method:

1. Why is my first recurrence relation not going to encompass both the even and odd parts if $$a_{n+2} \ne 0$$ for starting at both $$a_1$$ and $$a_0$$?
2. Is there some non-arbitrary need that if $$c=0 \implies a_1 = 0$$?
3. If my answer may only contain odd or even powers of $$x$$ but $$a_{n+2} \ne 0$$ for starting at both $$a_1$$ and $$a_0$$, then how do I know if I only have an odd or even power of $$x$$ for my recurrence relation?
4. Generally, what are the parametrisations of $$c$$ such that the solution will only encompass odd, even, or both powers of $$x$$?