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Bessel Generating Function

  1. Nov 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely,

    G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))},

    Hint. Starting from the recurrence relation

    Jn+1(x) + Jn-1(x) = (2n/x)Jn(x),

    show that G(x,t) satisfies the differential equation (t+1/t)G(x,t) = (2t/x)∂G/∂t. Partially integrate this equation and fix the unknown function of x by the requirement that the coefficient of t0 be J0(x)

    3. The attempt at a solution
    This is the setup I have so far. I'm not sure if this is the correct way to go. If it is please let me know so I can continue down the pipe line.

    ∫xe^((x/2)(t-1/t))∂t = ∫((2t^2)/(t^2)+1)∂G

    Any guidance would be helpful.
     
  2. jcsd
  3. Nov 19, 2015 #2
    Thanks for the post! 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?
     
  4. Nov 19, 2015 #3
    Thank you, Greg. You can bump this post.
     
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