Suppose we have the Generalized Legendre-type polynomial sequence $f_n(x)=\dfrac{d^n}{dx^n}(p(x)^n)$ , where $p(x)$ is any polynomials of degree at least $2$ and has at least two terms.
$1.$ Do these polynomial sequences satisfy any homogeneous linear ODE of polynomial coefficients? Find the one of minimal order if this is the case.
$2.$ Are these polynomial sequences orthogonal, especially when $p(x)$ are polynomials of degree more than $2$ ?