
> 
> **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](https://arxiv.org/pdf/2005.09809.pdf) 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.



Iterated antiderivatives of polynomials having many real roots
The question seems to be answered at the "Untitled" link from canvas.wisc.edu when you do a google search on "appell sequence real zeros".


