2001-32.md

course	course_year	question_number	tags	title	year
Methods	IB	32	IB 2001 Methods	4.I.2H	2001

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

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

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