Consider a Maclaurin series solution

$$ y = (1-x ^ 2) y '' -2xy '+ α (α + 1) y = 0, -1 <x <1. $$

CA watch $$ a_2 = frac {-α (α + 1)} {6} a_0 $$ $$ a_3 = frac {- (α – 1) (α + 2)} {6} a_1 $$

and for all $ n≥2 $,

$$ a_ {n + 2} = frac {n (n + 1) – α (α + 1)} {(n + 2) (n + 1)} a_n = frac {(n – α) (n + α + 1)} {(n + 2) (n + 1)} a_n. $$

Deduce this, if $ α = k ∈ {0,1,2,3, … }, $ then

$$ a_ {k + 2} = a_ {k + 4} = a_ {k + 6} = … = 0. $$

Therefore, write a polynomial solution of the Legendre equation in the cases $ α = $ 0,1,2,3,4.

For $ α = $ 3, write the first 4 terms of the solution of the Legendre equation in the other series and try to find a general expression for the coefficients of this series.