Series Expansion of Bessel Function of the First Kind/Negative Index

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

Then:

where $n$ is not a (strictly) positive integer.

Proof
From Series Expansion of Bessel Function of the First Kind:

The result follows by substituting $-n$ for $n$ in $1$ and simplifying.

Also see

 * Bessel Function of the First Kind of Negative Integer Order for when $n \in \set {-1, -2, -3, \ldots}$