special functions - modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder

$R_{k,m} equiv H_{k} ~ mod H_m$

for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial (orthogonal under the weight $w(x) = e^{-x^2}$) and $deg R_{k,m} leq m-1$. I haven't been able to find anything online, neither could compute it through the recurrence relation of Hermite polynomials...

Update:

The motivation for my question is as follows. The $m$-point Gauss-quadrature is obtained by placing the nodes at the roots of $H_m$ and choosing the weights accordingly such that integrating any polynomial (with respect to weight $w$) of order $leq 2m-1$ is exact. Now I want to know the error formula for polynomials of degree $k geq m$, especially $H_k$. By computing $H_k$ modulo $H_m$, the integration error is given by the integration of the remainder $R_{k,m}$.