Given the following Legendre polynomials, find the roots of $p_{2}$
$$p_{i}(x)=\frac{1}{2^{i}i!}\frac{d^{i}}{dx^{i}}[(x^2-1)^{i}]$$
My book says that $p_{2}(x)=x^2$ and can be adjusted to $p_{2}(x)=x^2+c$
But wikipedia says that $p_{2}$ is defined from a recursive property as $\frac{1}{2}(3x^2-1)$. So I am a little bit confused about what roots I should find. The other thing would be to plug in $i=2$ and then find the roots, but I am not sure about that method as well. Any explanation would be helpful
In order to evaluate $$p_{i}(x)=\frac{1}{2^{i}i!}\frac{d^{i}}{dx^{i}}[(x^2-1)^{i}]$$ for $i=2$, we differentiate $$(x^2-1)^2$$ twice to get
$$ 12x^2-4$$ and divide by $8$.
This is the correct answer$$\frac{1}{2}(3x^2-1)$$
The roots are $$x= \pm \frac{1}{\sqrt 3}$$