I'm looking for solutions to the Legendre differential equation

$$ \frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0 $$

with boundary conditions $f'(0)=0$ and $f(1)=0$, but with $\alpha$ free to choose (note that $\alpha$ is shifted by 1 compared to the usual parameter $l$ in Legendre's differential equation).

The standard power series approach yields a the usual recurrent relation for the terms, where it immediately follows from the condition $f'(0)=0$ that the series only contains even order terms:

$$ f(x)=\sum_{k=0}^\infty\frac{(-1)^k}{(2k)!}\left(\prod_{j=1}^k(2(j-1)+\alpha)(\alpha-2j+1)\right)x^{2k} $$

My question: does there exist values of $\alpha$ such that this boundary value problem has a non-trivial solution?


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