I want to calculate the Bessel function, given by
$$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$
I know there are some tables that exist for this, but I want to keep the $\beta$ variable (i.e. I want a symbolic form in terms of $\beta$). If there is a way to simplify the summation part of the equation and leave an equation only in terms of $\beta$, that would be very helpful. (I see there is a dependence on $2m$, but I would like to see a way to break down the "other half" of the equation.)
Another question I have is: how is this calculated for $\beta$ values that are greater than $1$? It seems to me that this would give an infinite sum.
I am looking for something for $\alpha=1,3,5$ and $\beta=4$.
Thanks in advance.