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I was using Mathematica to get the series solution for Legendre equation. But when I get the recurrence relation and use RSolve:

RSolve[Subscript[a, n + 2] == ((n - l) (n + l + 1))/((1 + n) (2 + n)) Subscript[a, n], 
   Subscript[a, n], n] // FullSimplify // TraditionalForm

(where l and n are both non-negative)

Just for calculating the coefficient of the series, I only find the solution is strange:

{{Subscript[a, n]->Γ(1/2 (-l+n))/Γ(-(l/2)) (2^n (Subscript[c, 2] (-1)^n+Subscript[c, 1]) Γ(1/2 (l+n+1)))/(Γ(n+1) Γ(1/2 (l+1)))}}

I was confused that when l and n are even and n < l,the answer will be ComplexInfinity. But it is so counter-intuitive!

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