Question For which polynomials $p_n:\mathbb{R} \rightarrow \mathbb{R}$ having $n$ distinct real roots can we find an infinite sequence of polynomials $$ p_n, p_{n+1}, p_{n+2} , p_{n+3}, \dots, $$ such that $p_i$ is a polynomial of degree $i$ with $i$ distinct real roots and $p_{i+1}$ is an anti-derivative of $p_i$ for all $i \geq n$?
Hermite polynomials have this property. The $n-$th Hermite polynomial $H_n$ has $n$ distinct roots and satisfies the derivative relation $$ \frac{d}{dx} H_n = n H_{n-1}.$$ This means that when computing the antiderivative of $H_{n}$, one can always find the correct constant to turn the anti-derivative into a multiple of $H_{n+1}$ and multiples of $H_{n+1}$ have exactly $n+1$ roots. My gut feeling would be that this is a very special property that not many polynomials have and I am wondering whether there is any sort of result in that direction.
Motivation This question is motivated by the asymptotic behavior of roots of polynomials when polynomials are differentiated many times. A while back I proposed a PDE to describe this and the PDE seems to be smoothing, so there should be a loss of information as one differentiates. More recently, Jeremy Hoskins and I proved a result that also hints towards loss of information. This sort of suggests that having these infinite chains should be somehow `rare' in some sense but it's more a vague connection than anything theorem-based.
