I recently came across the general form of Bessel Functions of half-integer order given by: $$ J_{n+\frac{1}{2}}(x)=(-1)^n\sqrt{\frac{2}{\pi}}x^{n+\frac{1}{2}}\left(x^{-1}\frac{d}{dx}\right)^n\frac{\sin{x}}{x}. $$ I am required to prove this using the recurrence relation: $$ J_{s\pm 1}(x)=\frac{s}{x}J_s(x)\mp J'_s(x). $$ I tried to prove by induction, but I think my main issue is that I am not sure what $\left(x^{-1}\frac{d}{dx}\right)^n\frac{\sin{x}}{x}$ actually means. I was thinking that the power $n$ is distributed across to $x^{-1}$ and $\frac{d}{dx}$ but that gives $x^{-n}$ which allows me to simplify the general form to be $$ J_{n+\frac{1}{2}}(x)=(-1)^n\sqrt{\frac{2}{\pi}}x^{\frac{1}{2}}\frac{d^n}{dx^n}\frac{\sin{x}}{x}. $$ If that is indeed true, textbooks should have presented this formula instead. I have used MATLAB to help me find the zeros using 2 different methods (the formula above, and by besselj) but gave me different answers. That is how I know that my simplification is wrong.
Could anyone kindly enlighten me what $\left(x^{-1}\frac{d}{dx}\right)^n\frac{\sin{x}}{x}$ means and how I can go about proving the general form using the recurrence relations? Any form of help is deeply appreciated! Thank you!
It means that you differentiate with regard to x, and then divide by x, and then repeat the entire process $($by repeatedly applying these two operations, in this exact order$)$, a total number of n times.