Solution
= Solution
Insert the <power series> and equate the coefficient of $x^n$. In the <Legendre differential equation> this gives
$$
(n+2)(n+1)a_{n+2}+[\ell(\ell+1)-n(n+1)]a_n=0.
$$
Hence the <recurrence relation> is
$$
\boxed{a_{n+2}=\frac{(n-\ell)(n+\ell+1)}{(n+1)(n+2)}a_n,\qquad n\geq0.}
$$
The free coefficients $a_0,a_1$ generate the independent even and odd series, respectively. They converge at least for $|x|<1$, up to the nearest singular points of the equation.