Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Closed form expression of the roots of Laguerre polynomials

  1. Apr 17, 2013 #1
    The Laguerre polynomials,

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

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

    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. jcsd
  3. Apr 17, 2013 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Closed form expression of the roots of Laguerre polynomials
Loading...