Derive recurrence relations for Bessel functions from the generating function

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Your problem describtion gives a good hint:

Calculate the Derive-ative of $G(x,h)$ on the LHS and your Bessel sum on the RHS!

$$\frac d{dx} \exp\left\{\frac x2\left(h-\frac1h\right) \right\} =\frac 12\left(h-\frac1h\right) \sum_{-\infty}^\infty h^n J_n(x)=\frac 12 \left(\sum_{-\infty}^\infty h^{n} J_{n-1}(x) - \sum_{-\infty}^\infty h^{n} J_{n+1}(x)\right)\\\frac d{dx}\sum_{-\infty}^\infty h^n J_n(x)=\sum_{-\infty}^\infty h^n\frac d{dx} J_n(x)$$ Now compare entries with equal $h^n$...

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Amy
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Updated on August 01, 2022

Comments

  • Amy
    Amy over 1 year

    I'm going through practice questions for my exams but this question has left me confused:

    The Bessel functions of integer order, $J_n(x)$, are described by the generating function:

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    Derive the recurrence relations.

    The markscheme has put these two as the answer

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    but has not showed any working, so I don't understand what I should have done to get there

  • Amy
    Amy over 9 years
    Hi thanks that's really helpful, but once I have the derivatives of LHS = derivates of RHS how do I get rid of the h^n terms and summation signs? that way I will get the second recurrence relation I wrote in the question