Legendre-Fourier series for $x^n$

Question

What is the full Legendre-Fourier series for $x^n$? I realize that this depends upon if $n$ is odd or even.

Progress

I wrote out the first three coefficients for the series, which are dependent upon whether $n$ is odd or even. I am obtaining the Legendre Polynomials themselves from Wikipedia.

Answer 1

Hint: Use the Rodrigues formula for Legendre polynomials and integration by parts in order to evaluate $$ \int_{-1}^{1} x^n P_m(x)\,dx. $$ Recall that $ \int_{0}^{1} x^k (1-x^2)^j\,dx $ depends on values of the Beta function. You should get:

$$ \int_{-1}^{1} x^n P_m(x)\,dx = (-1)^m\frac{1+(-1)^n}{n+m+1}\prod_{k=1}^{\lfloor m/2\rfloor}\frac{n-m+2+2k}{n+m+1-2k}.$$

Answer 2

I just want to mention that you got some errors in your formula...

The most easy way to see it is by noticing that your solution equals $0$ for all odd $n$ due to the prefactor $\frac{1+(-1)^n}{n+m+1}$.

I would suggest the modification:

\begin{equation} c_m =\int \limits_{-1}^1 x^n P_m(x) dx = \frac{1+(-1)^{n+m}}{n+m+1} \prod\limits_{k=1}^{\left\lfloor m/2\right\rfloor } \frac{n-m+2k}{n+m+1-2 k} \end{equation}

which gives the correct expansion-coefficients for

\begin{equation} x^n = \sum\limits_{m=0}^n \frac{2m+1}{2} ~ c_m P_m(x) \end{equation}