Derive recurrence relations for Bessel functions from the generating function

Question

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

Answer 1

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 $h^n$...