For some orthogonal polynomials, their roots can be expressed in closed form. For exemple, for the Chebychev polynomials of the second kind:
$$
U_n(x) = \frac{\sin((n+1)\arccos(x))}{\sin(\arccos(x))}
$$
the roots are, for any order $ n $: $$ x_k^{(n)} = \cos\left(\frac{k\pi}{n+1}\right)$$
However, for Hermite polynomials $ H_n $ there seem not to be a closed-form expression and numerical computation must be invoked (e.g. Newton's method)…
Why is it so?
I have looked at Abramowitz and Stegun, section 26.16 as well as table 25.10 with the numerical values of the roots of $H_n$ (and the reference therein by Salzer, Zucker and Capuano, available at https://nvlpubs.nist.gov/nistpubs/jres/048/jresv48n2p111_A1b.pdf).
Best Answer
In general there should be no reason to expect closed form radical expressions for the roots of polynomials with degrees greater than 5, much less general formulas. Note for example that there is no way to reduce $\cos(k\pi/n)$ to radicals for arbitrarily $n$.
And it is certainly the general case that one should not expects roots to be given in terms of things other than radicals, unless the definition clearly motivates such like with the Chebyshev polynomials. Since there is no such form for Hermite polynomials which makes roots explicit, one cannot expect a closed form to exist.