A generalization of the Hermite polynomial: is there a name for this class of polynomials in the literature?

Question

An explicit expression for the `probabilist's' Hermite polynomial is given by $$\sum_{r=0}^{\lfloor n/2\rfloor}(-1)^r\frac{n!}{2^r(n-2r)!r!}x^{n-2r}.$$ In playing around with some combinatorics, I came across the following variant $$\sum_{r=0}^{\lfloor n/k\rfloor}(-1)^r\frac{n!}{(k!)^r(n-kr)!r!}x^{n-kr}.$$ Do these polynomials belong to any known or standard class of (orthogonal) polynomials? Is there anything known about them?

Answer 1

So i went over some literature, and you can express your polynomials as $$_SH_n^{(k)}(x,y)=\sum _{r=0}^{n/k}\frac{n!y^rx^{n-rk}}{r!(n-rk)!},$$ by doing $y=\frac{(-1)}{k!}.$

This polynomials are called Gould Hopper. For example check the paper(Table I, first row):

https://arxiv.org/pdf/1911.09139.pdf