I'm staring at the Bessel functions, and I'm trying to figure out exactly why $J_0$ should behave differently than $J_{1^+}$: at zero, only $J_0(0)=1$, and $J_{n\geq1}(0)=0$.
Even all of $Y_n$ behave in the same singular way:
It strikes me as odd that one element in an infinite series would exhibit unique behavior at 0. It seems like there should be an explanation for the behavior of $J_0$ other than "it solves the equation," but I'm not sure what that might be.
Is there one?
Another way to derive it is from the representation of $J_n(x)$ as Fourier series coefficients: $$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-int} e^{ix \sin(t)}\; dt $$
So $$J_n(0) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-int}\; dt $$ which of course is $0 $ for integers $n \ne 0$ and $1$ for $n=0$.