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Legendre Polynomials & the Generating function

  1. Nov 3, 2016 #1
    1. The problem statement, all variables and given/known data
    upload_2016-11-3_23-24-38.png
    2. Relevant equations
    upload_2016-11-3_23-24-49.png
    and in chapter 1 I believe that wanted me to note that
    upload_2016-11-3_23-35-14.png
    3. The attempt at a solution
    For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre polynomial will be odd. With odd Legendre polynomials, every coefficient has a term of x attributed to it (example P3(x) = 1/2(5x^3 - 3x)), so if it were to be evaluated at 0, the result will always be zero. However I am not sure how to prove that using the Generating function.

    As for the second part of this question, I am not really sure what to do. For now I am looking for a place to start with that.

    [EDIT] I did expand 5.1 as a Maclaurin Series, but I don't see how they want me to equate that to 5.2.
     

    Attached Files:

    Last edited: Nov 3, 2016
  2. jcsd
  3. Nov 9, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Nov 13, 2016 #3
    I figured out the solution, just posting it here for sake of completeness.

    upload_2016-11-13_21-29-28.png
     
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