Originally Posted by
Chris L T521 You need to mess around with the series for the Bessel function of the first kind:
$\displaystyle J_n(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m+n}}{2^{2m+n}m!\Gamma(m+n+1)}$
Clearly,
$\displaystyle J_{-n}(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)}$
Now, make a substitution. Let $\displaystyle m=m^{\prime}+n$. Then,
$\displaystyle \begin{aligned}\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)} &=\sum\limits_{m^{\prime}+n=0}^{\infty}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p rime}+n+1)}\\ &= \sum\limits_{m^{\prime}=-n}^{-1}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p rime}+n+1)}+\sum\limits_{m^{\prime}=0}^{\infty}\df rac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p rime}+n+1)}\end{aligned}$
A couple observations need to be made to finish it up.
Can you proceed?