Chapter 11.7, Problem 14E

a. Use formula (25) to show that $P_n\left(x\right)$ is an odd function when $n$ is odd and is an even function when $n$ is even.

b. Show that the odd degree Legendre polynomials are orthogonal on $\left(0,1\right)$; that is,

${\displaystyle {\int }_0^1{P}_{2m+1}\left(x\right)P_{2m+1}\left(x\right)dx}=0,m\ne n$.

c. Show that the even degree Legendre polynomials are orthogonal on $\left(0,1\right)$; that is,

${\displaystyle {\int }_0^1{P}_{2m}\left(x\right)P_{2n}\left(x\right)dx}=0,m\ne n$.

$P_n\left(x\right)=2^{-n}{\displaystyle \underset{m=0}{\overset{\left[n/2\right]}{\sum }}\frac{{\left(-1\right)}^m\left(2n-2m\right)!}{\left(n-m\right)!m!\left(n-2m\right)!}{x}^{n-2m}}$ … $\left(25\right)$