I have a few questions about the Bessel function of the first kind at its derivatives. Please excuse my possible lack of rigour as I am a Physics student.
For a project, I have to solve Bessel's differential equation with the form:
$$ r^2 \frac{\partial^2 R_{(r)}}{\partial r^2} + r \frac{\partial R_{(r)}}{\partial r} + (k^2r^2 - n^2) R_{(r)} =0 \text{.}$$
This where my first question starts with the formalism for the shorthand of the Bessel function of the first kind because WolframAlpha and the articles I'm working with write the solution as some constant times $J_n (rk)$, where the Bessel function of the second kind is thrown out due to boundary condition $R_{(r)}$ can't go to infinity as $r \to 0$. Does this $J_n(rk)$ shorthand correspond to the following explicit expression?
$$ J_n (rk) = \sum_{m=1}^{\infty} \frac{(-1)^m}{m!(m+n)!}\Big(\frac{rk}{2}\Big)^{2m +n} $$
My following question relates to the partial derivative of $J_n(rk)$ and if it follows the following identity,
$$ \frac{\partial}{\partial r} J_n (rk) = J_{n-1} (rk) - J_{n+1} (rk)$$
where the explicit expression would then be:
$$ J_{n-1} (rk) - J_{n+1} (rk) \\ = \sum_{m=1}^{\infty} \frac{(-1)^m}{m!(m+n-1)!}\Big(\frac{rk}{2}\Big)^{2m +n-1} - \sum_{m=1}^{\infty} \frac{(-1)^m}{m!(m+n+1)!}\Big(\frac{rk}{2}\Big)^{2m +n+1} \text{.} $$
I would also like to note that $n$ is strictly an integer in this case in order to avoid multivalued points and $k$ is free to be any real number (although I could be wrong on this).