Simplify this expression that came from integration

Question

I was doing a calculation and arrived at a term $\left[P_{l-1}(\cos(\theta)) -P_{l+1}(\cos(\theta))\right]_{0}^{\pi}$(So this is the result of an integration). Does anybody of you know how to simplify this expression? (Testing suggested to me that this one is either $0$ or $2$, but I did not get there). $P_n$ is the $n$-th Legendre polynomial.

Answer 1

Two things:

• Legendre polynomial $P_{l}(x)$ is either even (for even $l$) or odd (for odd $l$) function of $x$,

• Its value at $x=1$ is $P_l(1)=1$ (this is a part of its definition).

I think you can get the answer from there.

Answer 2

Note that $$P_n(x)=\frac{1}{2^n}\sum_{k=0}^n\binom{n}{k}^2(x-1)^{n-k}(x+1)^k$$ See here for details. Your question asks for evaluating $$P_{l-1}(-1)-P_{l+1}(-1)-P_{l-1}(1)+P_{l+1}(1)$$ Now, easily we see $$P_n(1)=1,\ P_n(-1)=(-1)^n$$ So the expression evaluates to $$(-1)^{l-1}-(-1)^{l+1}-1+1=0$$