Integrating Legendre polynomials

Question

I need to solve following integral: $I_{n}=\int_{-1}^{1}\frac{1}{x}P_{n}(x)P_{n-1}(x)dx$. I have hint that following equation needs to be used: $(n+1)I_{n+1}+nI_{n}=2$. Does anyone have idea how to proceed?

Answer 1

First of all, it is easy to see that $I_1 = 2$. Then, after proving the given recurrence relation, you can see that: $I_2 =0$, $I_3=\frac{2}{3}$, $I_4=0$, $I_5=\frac{2}{5}$, ... It is clear that you have ($n \geq 1$): $$I_{2n} =0$$ and $$I_{2n-1} = \frac{2}{2n-1}$$