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Math Help - Hermite polynomials

  1. #1
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    Hermite polynomials

    I need some help if possible with the following problems.The statements are long so please bare with me.
    This is concerned with a system of polynomials, the Hermite polynomials,
    which arise in various applications, in particular in the analysis of the normal distri-
    bution in probability and in the solution of certain di erential equations from physics.
    The project explores the properties of these polynomials, including the di erential
    equation they satisfy.The polynomials w are in fact a variant of the standard
    Hermite polynomials. This version is more convenient for the applications to
    probability, and di ffers from the usual version by a sign and a scaling of the x-variable.

    a)The polynomials are defined as follows H_n(x) for n = 0; 1; : : : as follows: fi rst, set H_0(x) =1 and H_1(x) = -x; then, for n  2, H_n is defined by the recurrence
    H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x): (1)
    I have to use (1) to verify that H_2(x) = x^2 -1 and H_3(x) = 3x-x^3, and calculate H_4(x)and H_5(x).
    I also have to show that that H_n is an even function when n is even, and that it is an odd function when n is odd (maybe using (1) and induction on n).
    Also that also (maybe I should still use induction and (1)) that
    H_2k(0) = (-1)^k(2k -1)(2k -3) ....1:
    What is the value of Hn(0) when n is odd?

    b)Here I must show that t H_n satis es a di erential equation. By di erentiating (1) and
    using induction on n, show that, for n >= 1,
    H'_n(x) = -nH_(n-1)(x) (2)
    I have to (2) to express H_(n-1) and H_(n-2) in terms of derivatives of H_n, and substitute these into (1) to show that
    H''_n - xH'_n + nH_n = 0 (3)
    for n>= 0. Now let O_n(x) = exp(-(x^2)/4 )H_n(x). Use (3) to show that
    O''_n +(n+1/2-(x^2)/3)O_n=0
    Thank you.
    
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  2. #2
    A Plied Mathematician
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    Is this for a project you're handing in?
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  3. #3
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    Yes...I'm trying to get a head start...I have(and had) a few ideas hope they will work(this time),I'l be posting my progress if that's ok....any advice?
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  4. #4
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    Quote Originally Posted by AkilMAI View Post
    Yes...I'm trying to get a head start...I have(and had) a few ideas hope they will work(this time),I'l be posting my progress if that's ok....any advice?
    Are you the same person as Houdini?
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  5. #5
    A Plied Mathematician
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    If the project is for a grade, then it is forum policy not knowingly to help with such work. See Rule # 6 here.
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  6. #6
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    I apologize....I posted in the wrong part forum.No,he is my roomate.My answer lies in 2nd order DE constant coefficients and boundary conditions
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by AkilMAI View Post
    I apologize....I posted in the wrong part forum.No,he is my roomate.My answer lies in 2nd order DE constant coefficients and boundary conditions
    Then why are you answering questions addressed to Houdini (not once but at least twice)?

    You are not getting the benefit of the doubt, used ID Houdini has been banned.

    CB
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