I've been reviewing Legendre Polynomials for problems in electrostatics recently, and I came across something that I hadn't thought of the first time. So for the function to be well behaved on the interval $-1 \leq x \leq 1$, we want the series with the coefficients:
$$ a_{n+2} = \frac{n(n+1)-l(l+1)}{(n+1)(n+2)} a_n $$
to terminate for some $n$. The first, most obvious solution is that the series terminates when $n=l$ if $l$ is a positive integer. However if we send $l \rightarrow -(l+1)$, then the equation remains unchanged (which makes sense since this is the other root to the equation $n(n+1) - l(l+1) = 0$). Why then does every author (that I've encountered or been able to find) say that $l$ must be a positive integer? Is it just because using $-(l+1)$ would be equivalent to using negative integers as well so we choose to restrict ourselves to the positive integers? Thanks in advance for any answers!