Derive recurrence relations for Bessel functions from the generating function
Your problem describtion gives a good hint:
Calculate the Deriveative of $G(x,h)$ on the LHS and your Bessel sum on the RHS!
$$\frac d{dx} \exp\left\{\frac x2\left(h\frac1h\right) \right\} =\frac 12\left(h\frac1h\right) \sum_{\infty}^\infty h^n J_n(x)=\frac 12 \left(\sum_{\infty}^\infty h^{n} J_{n1}(x)  \sum_{\infty}^\infty h^{n} J_{n+1}(x)\right)\\\frac d{dx}\sum_{\infty}^\infty h^n J_n(x)=\sum_{\infty}^\infty h^n\frac d{dx} J_n(x)$$ Now compare entries with equal $h^n$...
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Amy
Updated on August 01, 2022Comments

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:
Derive the recurrence relations.
The markscheme has put these two as the answer
but has not showed any working, so I don't understand what I should have done to get there

Amy over 9 yearsHi 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