Can someone help me to prove this propertie of Legendre Polynomials?

User13587

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Aug 27, 2022
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I already prove other properties of the Legendre polynomials, like:

Pn(x)=(1)nPn(x)P_n(-x) = (-1)^nP_n(x)P2n+1(0)=0P_{2n+1}(0) = 0Pn(±1)=(±1)nP_n(\pm1)= (\pm1)^n
But I can't get this:

P2n(0)=(1)n(2n1)!22n1((n1)!)2P_{2n}(0) = \frac{(-1)^n(2n-1)!}{2^{2n-1}((n-1)!)^2}

I'm so glad if someone can help me with this.
 
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