Let $P_i:[-1,1]\to\mathbb{R}$ be the degree $i$ polynomial of a family of orthogonal polynomials for which the following conditions hold:
- The root of $P_1$ is $0$.
- When $n\geq2$, the roots $x_1,\ldots ,x_n$ of $P_n$ satisfies
$$\frac{x_i}{x_n}=\sin \left(\pi \left(\frac{i-1}{n-1}-\frac{1}{2}\right)\right)$$
How to determine which family of orthogonal polynomials this is?
Edit After choosing $P_0(x)=1,P_1(x)=x$, numerical experiments reveals that $P_2$ can be chosen as any, even quadratic polynomial. The recurrence relation for the rest is $$P_n(x)=xP_{n-1}(x)-P_{n-2}\begin{cases} 0 & n=3 \\ c & \text{else} \end{cases}$$
where $c$ doesn't depend on $n$. Because of the $0$ the roots of $P_2$ are also roots of the remaining $P$s, so no family of orthogonal polynomials exist.