Results 1 to 3 of 3

Math Help - help solving orthogonal polynomial

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    39

    help solving orthogonal polynomial



    I managed to do the first part, stuck in the part circled. Any help will be appreciated, thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Use the Rodrigues formula plus the following identity:

    H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x).

    Does that give you some ideas?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,373
    Thanks
    48
    Use your generating function

    \displaystyle  e^{2xu - u^2} = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H_n(x)

    Then differentiating wrt x gives

    \displaystyle 2u e^{2xu - u^2} = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x)

    so using the generating function again gives

    \displaystyle  2u \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H_n(x) = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x)

    or

    \displaystyle \sum_{n=0}^{\infty} \dfrac{u^{n+1}}{n!}2H_n(x) = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x)

    Now, shifting indicies gives

    \displaystyle \sum_{n=1}^{\infty} \dfrac{u^{n}}{(n-1)!}2H_{n-1}(x) = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x)

    or

    \displaystyle \sum_{n=1}^{\infty} \dfrac{u^{n}}{n!}2nH_n(x) = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x).

    Thus,

    \displaystyle \sum_{n=0}^{\infty} \dfrac{u^{n}}{n!}2nH_n(x) = \sum_{n=0}^{\infty} \dfrac{u^n}{n!}H'_n(x)

    since the first term of the series on the left is zero. Equating coefficients of u gives your result.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving a Polynomial Equation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 20th 2011, 05:52 PM
  2. Replies: 1
    Last Post: August 15th 2011, 04:32 AM
  3. Solving Polynomial Expressions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 13th 2008, 07:56 AM
  4. solving polynomial
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 28th 2008, 07:42 PM
  5. Plzz Help solving polynomial
    Posted in the Algebra Forum
    Replies: 6
    Last Post: March 22nd 2007, 08:20 PM

Search Tags


/mathhelpforum @mathhelpforum