In a problem in QM I faced with this polynomials, sadly they are not orthogonal. I was wondering if someone else knows these polynomials, I was looking up and didn't find anything about them but they are similar to Hermite's Polynomials
The polynomials of order $2p+1$ are given by,
$$ P(x)=x \prod_{i=1}^{P}(-4i+x^2) $$
This polynomial was the result of finding the eigenvalues of a combination of some matrices.
My question is: Is there a way to express this polynomial as some other orthogonal polynomials?
I need an orthogonal form of $P(x)$ in order to find orthogonal eigenvectors of my matrices.