Properties about the Generalized Hermite-type polynomial sequence
$f_n(x)=e^{-p(x)}\dfrac{d^n}{dx^n}e^{p(x)}$
Suppose we have the Generalized Hermite-type polynomial sequences
$f_n(x)=e^{-p(x)}\dfrac{d^n}{dx^n}e^{p(x)}$ , where $p(x)$ is any
polynomials of degree at least $2$ .
$1.$ Do these polynomial sequences satisfy any 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 $2$ ?