The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the Skellam distribution, and the von Mises distribution). This is its formula (NB: this is false...see answer below):
$$ J_k(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+k+1)}\left(\frac{x}{2}\right)^{2m+k} $$
How can we simplify this thing? Because I am interested in Skellam distributions in particular, I am interested in the case where $k$ is a possibly-negative integer, and $x$ is positive.
This thing appears to depend on evaluations of the Gamma function at negative values, which can take on the value complex infinity. However, the wiki page for the Skellam distribution mentions that the Bessel function with a negative integer $k$ will be equal to the Bessel function with the absolute value of that integer as a subscript. Why is this?