I'm trying to get the recurrence relation $$ J_{\nu-1}-J_{\nu+1}= \frac{2\nu}{x}J_\nu$$ using the series representation of the Bessel function of the first kind $$ J_\nu = \sum\limits_{k=0}^\infty \frac{(-1)^k }{k!\Gamma(\nu+2k+1)}\left(\frac{x}{2}\right)^{\nu+2k}$$ but I am getting stuck at $$J_{\nu-1}-J_{\nu+1} = \sum\limits_{k=0}^\infty \frac{(-1)^k }{k!\Gamma(\nu+2k+1)}\left(\frac{x}{2}\right)^{\nu+2k}\left[\left(\frac{x}{2}\right)^{-1}(\nu+2k)-\left(\frac{x}{2}\right)\frac{1}{\nu+2k+1}\right]$$ I know that there are other ways to get the recurrence relation, but it should be also possible to get it using this method, or am I wrong? Can someone tell me what I am doing wrong ?
Thanks!
I just stumbled accross the answer of my question in a book about Special Functions. So here's the step-by-step solution.