Question on Legendre equation

Question

I have a doubt. If Legendre equation has a polynomial solution, is the constant $l$ in $l(l+1)$ necessarily a integer number? Asked in another way, is possible $l(l+1)$ be a integer if $l$ is not an integer?

Thanks in advance.

Answer 1

Yes (although this is not really relevant for the Legendre polynomials): $$l = \frac{-1 + \sqrt{5}}{2}$$ $$l(l+1) = 1$$