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:

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

Proof

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Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {II}$. Bessel functions: $4$