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Orthogonality of Associated Laguerre Polynomial

  1. Mar 17, 2015 #1
    I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial :

    upload_2015-3-17_17-49-19.png
    to mutual orthogonality equation :

    upload_2015-3-17_17-49-40.png

    and set, first for upload_2015-3-17_17-50-7.png and second for upload_2015-3-17_17-50-20.png .

    But after some step, I get trouble with this stuff :

    upload_2015-3-17_17-50-31.png

    I've already search solution for this form but still no light. Any body here could help?
     
  2. jcsd
  3. Mar 17, 2015 #2

    ShayanJ

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    The best way is using the differential equation. Write it for ## L^k_n ## and ## L^k_m ##. Then multiply the first by ## L^k_m## and the second by ## L^k_n ##. Then subtract one from the other. Come back if you encounter a problem!
     
    Last edited: Mar 17, 2015
  4. Mar 17, 2015 #3
    Well I still didn't get it. ## L^k_n ## has a rodrigues form and also associated Laguerre polynomial. Which one I need to use?
     
  5. Mar 17, 2015 #4

    ShayanJ

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    I said the differential equation, which is:
    ## x \frac{d^2 L_n^{k}}{dx^2} +(k+1-x) \frac{dL_n^{k}}{dx} +n L_n^k =0##
     
  6. Mar 22, 2015 #5
    Nah... I got it. But still, I am stuck when connecting it with the orthogonality....
     
  7. Mar 23, 2015 #6

    ShayanJ

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    I explained it in that post! Just write the equation two times for ## L_m^k ## and ## L_n^k##. Then multiply the first by ## L_n^k## and the second by ## L_m^k ## and subtract one from the other. Then find an integrating factor and multiply the equation by it. You'll find out how to continue but come back if you didn't.
     
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