I am not familar with Bessel functions, however I know they are derived from the differencial equation,
We realize that, is a regular singular point.
Then, we use a theorem about regular singular linear second order homogenous differencial equations. You have to look at the radii of convergence of the two analytic functions in,
Then an analytic solution exists and its radii of convergence is as least as large as of .
Basically what I am trying to say is that there is a theorem that gaurenttes a solution with a certain radii of convergence.
In the case of Bessel functions,
The two functions,
are analytic with radii of convergence the entire number line. Thus there exists an analytic solution (Bessel function) whose radii of convergence is at least as large as the other two (that is the entire number line).
Does that help at all? This is a subject I am not well versed, but I think knowing that theorem might help.