Formula_5370

\[\begin{gathered}(n+1) P_{n+1}(x)=(2 n+1) x P_{n}(x)-n P_{n-1}(x), \\P_{n}(-x)=(-1)^{n} P_{n}(x), P_{n}(1)=1, P_{n}(-1)=(-1)^{n}, \\\left(1-x^{2}\right) P_{n}^{\prime}(x)=n P_{n-1}(x)-x n P_{n}(x), \\P_{n+1}^{\prime}(x)-P_{n-1}^{\prime}(x)=(2 n+1) P_{n}(x) .\end{gathered}\]