The generating function of the Bessel function can be written as:
$$g_{(x,t)} =\sum_{r=0}^{\infty}\sum_{s=0}^{\infty} (-1)^s (\frac{x} {2})^{r+s} \frac{t^{r-s}} {r!s!} $$
Changing the summation index $r$ to $n=r-s$ leads to:
$$g_{(x,t)} =\sum_{n=-\infty}^{\infty}[\sum_{s=n}^{\infty} \frac{(-1)^s } {(n+s)!s!} (\frac{x} {2})^{n+2s}] t^n$$
How the new summation bounds are obtained? I can understand the variables and factorials' new form in this new summation index, but can not figure out the range of the converted double summation. Thank you for your help.