The generating function states that $$e^\left (\frac12 \left (t - \frac1t \right ) x\right )= \sum_{n = -\infty}^{\infty} J_n(x) t^n.$$ However, I do not know where the $n = -\infty$ came from in
.
Can someone point me in the right direction?
2026-04-02 20:30:19.1775161819
Prove the generating function for the Bessel function.
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A typical term after the multiplication is:
$$\frac{(-1)^k(\frac{x}{2})^{m+k}}{m!k!}z^{m-k}$$
The author then collects like-terms by 'level' $n=m-k$.