Obtain a first order DE for a Bessel function of order $n$ An exercise in my textbook is the following: 
Let $J_n (z)$ denote the Bessel function of order n. Set 
$$\theta(z,t)=\sum_{n=-\infty}^{+\infty}J_{n}(z)t^n$$
and assume that the series converges. 
Obtain a first order DE relating $\frac{\partial \theta}{\partial z}$ and $\theta(z,t)$.
I hope anybody can help, because I am totally stuck at this moment. 

$$\frac{\partial \theta}{\partial z}=\sum_m J'_m(z)t^m$$
Now using the derivative properties for the Bessel function, one has
\begin{equation}
\frac{\partial \theta}{\partial z}=\frac{1}{2}\sum \left[J_{m-1}(z)-J_{m+1}(z)\right]t^m
\end{equation}
It remains to shift the index $m$ in the summations to obtain
\begin{align}
\frac{\partial \theta}{\partial z}=\frac{1}{2}\left[t\sum_m J_m(z)t^m-\frac{1}{t}\sum_m J_m(z)t^m\right]\\
&=\frac{1}{2}\left( t-\frac{1}{t} \right)\theta(z,t)
\end{align}
Actually, $\theta$ is a generating function for the Bessel functions.
