Closed form expression of the roots of Laguerre polynomials

1. Apr 17, 2013

kd6ac

The Laguerre polynomials,

$L_n^{(\alpha)} = \frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x}x^{n+\alpha} \right)$

have $n$ real, strictly positive roots in the interval $\left( 0, n+\alpha+(n-1)\sqrt{n+\alpha} \right]$

I am interested in a closed form expression of these roots, that is, I would like to avoid any method of finding these roots, such as, Laguerre's method.

Any ideas are most welcome.

2. Apr 17, 2013

fzero

For a polynomial to be solvable by radicals, the Galois group of the polynomial must be solvable http://en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals. For degree 5 and above, there are many polynomials that are not solvable, so there is no closed form expression for the roots in terms of radicals. In rare examples, expressions for the roots in terms of special functions might exist.

It appears that the Laguerre polynomials are definitely not solvable (for example http://arxiv.org/abs/math/0406308). I haven't been able to turn up anything about special expressions for roots, so I'd guess that they don't exist.