Generating Function for Bessel Function of the First Kind of Order n of x

Theorem
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Then:
 * $\map \exp {\dfrac {x \paren {t - \frac 1 t} } 2} = \ds \sum_{n \mathop = -\infty}^\infty \map {J_n} x t^n$