Methods | Part IB, 2001

The Legendre polynomial Pn(x)P_{n}(x) satisfies

(1x2)Pn2xPn+n(n+1)Pn=0,n=0,1,,1x1.\left(1-x^{2}\right) P_{n}^{\prime \prime}-2 x P_{n}^{\prime}+n(n+1) P_{n}=0, \quad n=0,1, \ldots,-1 \leqslant x \leqslant 1 .

Show that Rn(x)=Pn(x)R_{n}(x)=P_{n}^{\prime}(x) obeys an equation which can be recast in Sturm-Liouville form and has the eigenvalue (n1)(n+2)(n-1)(n+2). What is the orthogonality relation for Rn(x),Rm(x)R_{n}(x), R_{m}(x) for nmn \neq m ?

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