1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Laguerre polynomials

  1. Oct 10, 2006 #1
    What differential equation does

    [tex]\phi_n (x) := e^{-x/2} L_n (x)[/tex]

    solve? [tex]L_n[/tex] is a Laguerre polynomial.

    Please give me a hint on this one. I haven't got a clue where to start.
  2. jcsd
  3. Oct 10, 2006 #2


    User Avatar
    Homework Helper

    What differential equation do the Laguerre polynomials solve? If you don't know, look it up.
  4. Oct 11, 2006 #3
    The Laguerre differential equation is

    [tex]xy'' + (1 - x)y' + ny = 0[/tex]

    and [tex]L_n (x)[/tex] is a solution to this but my [tex]\phi_n (x)[/tex] is not a solution to the Laguerre equation, is it?
    I know that [tex]\phi_n (x)[/tex] should solve a self adjoint differential equation but I don't think the Laguerre eq. is?
  5. Oct 11, 2006 #4


    User Avatar
    Homework Helper

    Find [itex]\phi',\phi''[/itex] in terms of the derivatives of L and use the differential equation relating the derivatives of L to get a DE relating the derivatives of [itex]\phi[/itex].
  6. Oct 11, 2006 #5
    Ok, so then I get

    [tex]x \phi_n^{''} (x) + (1-x) \phi_n^{'} + (n + \frac{1}{2} - \frac{x}{4}) \phi_n (x) = 0[/tex]

    but this isn't self-adjoint?
    In my case, self-adjoint means it can be written in the form

    [tex]\frac{d}{dx} (p(x) \frac{d}{dx} \phi_n (x) ) + q(x) \phi_n (x)[/tex]
  7. Oct 12, 2006 #6
    I made a mistake. The equation I get is

    [tex]x \phi_n^{''} + \phi_n^{'} + (n + \frac{1}{2} - \frac{x}{4}) \phi_n = 0[/tex]

    and this is indeed self-adjoint. Thanks for your help!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook