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Legendre polynomials proof question.Help

  1. Nov 16, 2006 #1
    Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could any one help me or give me some hints for this question?

    [​IMG]
     
  2. jcsd
  3. Nov 16, 2006 #2

    siddharth

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    Have you tried to solve it using the generating function identity?
     
  4. Nov 16, 2006 #3

    dextercioby

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    The first one is really easy. Just use the ODE whose fundamental solutions are Legendre's polynomials of the first & second kind.

    Daniel.
     
  5. Nov 16, 2006 #4

    dextercioby

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    For the second you'll need to perform this integral

    [tex] \int_{0}^{\pi} \left(\cos t \right)^{2n} \ dt \ , \ n\in \mathbb{N} [/tex]

    Daniel.
     
  6. Nov 16, 2006 #5
    could you give me detailed solution for understanding it better.
     
  7. Nov 17, 2006 #6
    sorry but i really got troubled with these questions.could anyone give me detail suggestions?
     
  8. Nov 17, 2006 #7
    Hello why noone give any hints?
    i tried also generating function.but i couldnt find.
    can anyone give me suggestion or show me a way to solve these problems?
     
  9. Nov 17, 2006 #8
  10. Nov 19, 2006 #9

    Meir Achuz

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    Use the generatling function. Expand g=[1+2tx+t^2]^{-1} in the binomial series. For the first case, use dg/dx for x=1.
    For the second case, use dg/dx for x=0.
     
  11. Nov 19, 2006 #10
    i tried generating function but i couldnt solve it.could you tell me step by step to solve this?
     
  12. Nov 20, 2006 #11

    siddharth

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    How about you post and tell us where you are stuck?

    Remember, the homework helpers here will help you solve the problem, not solve the problems for you.
     
  13. Nov 21, 2006 #12
    ok how can i send you my solution?
     
  14. Nov 21, 2006 #13

    siddharth

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    You can post it in this thread using LaTeX.
    Here's a tutorial on how it's used. You can click on any LaTeX image and see the code for the image.
     
  15. Nov 21, 2006 #14
  16. Nov 21, 2006 #15

    siddharth

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    For the first one, it looks like you've guessed the final result, rather than proven it.
    As Meir Achuz told you, use the generating function

    [tex] \frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^{\infty} P_n(x) t^n [/tex]

    So for the first one, differentiate wrt x, set x=1 and equate coefficients of t^n (using the binomial series expansion). Use a similar procedure for the second question.
     
  17. Nov 21, 2006 #16
    what does it meant wrt x?
     
  18. Nov 21, 2006 #17
    wrt x?????
     
  19. Nov 23, 2006 #18
    any help???????
     
  20. Nov 23, 2006 #19

    dextercioby

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    wrt= short for "with respect to". As for help, i think you've gotten enough ideas to solve the problem.

    I c a certain reluctance towards accepting advice. Maybe you shouldn't have come here in the first place. I hope that no one on this board will help someone by solving the problem entirely, because, to me, that's what you're asking for...

    Daniel.
     
  21. Nov 23, 2006 #20
    primitive polynomials in GF(4)
     
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