Legendre Polynomial definite integral identity

Question

I'm doing a problem involving legendre polynomials and I got stuck in this integral:

$$I_k=\int_{-1}^{1} x P_{2k+1}(x)dx $$

Update: Note that the function in the integral is even

If $k=0$, then $I_0=2/3$, but for the other values I don't know how to proceed

For integer $k>0$ I computed the integral in Wolfram(here is for k=1) and I obtained:

$$\int_{-1}^{1} x P_{2k+1}(x)dx=0 $$

Is this an identity for legendre polynomials? and if so, How can I prove it?

Answer 1

The Legendre polynomials $P_n(x)$ are solutions of ODE of the form $$\frac{d}{dx}\left[ A(x) \frac{dP_n(x)}{dx} \right] = \lambda_n P_n(x) \quad\text{ with }\quad \begin{cases} A(x) &= 1-x^2,\\ \lambda_n &= -n(n+1) \end{cases}$$

For any two distinct $n, m \ge 0$, notice $$\begin{align} & \frac{d}{dx} \left[ A(x) \left(P_n(x) \frac{dP_m(x)}{dx} - P_m(x) \frac{dP_n(x)}{dx}\right)\right]\\ = & P_n(x) \frac{d}{dx}\left[ A(x) \frac{dP_m(x)}{dx} \right] - P_m(x) \frac{d}{dx}\left[ A(x) \frac{dP_n(x)}{dx} \right]\\ = & (\lambda_m - \lambda_n) P_m(x)P_n(x)\end{align} $$ We have $$\int_{-1}^1 P_m(x)P_n(x) dx = \frac{1}{\lambda_m - \lambda_n} \left[ A(x) \left(P_n(x) \frac{dP_m(x)}{dx} - P_m(x) \frac{dP_n(x)}{dx}\right)\right]_{-1}^1 = 0$$ because $A(x) = 0$ at $x = \pm 1$.

Since $P_1(x) = x$, this means for any $n \ne 1$, we have $$\int_{-1}^1 xP_n(x) dx = 0$$