Simplification of a sum of Bessel functions

The proposed series can be converted into a generating function for the Bessel function by shifting the summation index ($n=p+k$): \begin{align} \alpha_p &= \sum_{n=-\infty}^{+\infty} (-i)^n J_{n-p}(R)e^{i(n-p)\Phi}\\ &= (-i)^p\sum_{k=-\infty}^{+\infty} (-i)^k J_k(R)e^{ik\Phi}\\ &=(-i)^p\sum_{k=-\infty}^{+\infty} J_k(R)e^{ik(\Phi-\pi/2)} \end{align} The generating function of the Bessel function reads \begin{equation} e^{\frac{1}{2}z(t-t^{-1})}=\sum_{k=-\infty}^{\infty}t^{k}J_{k}\left(z\right) \end{equation} It is valid for $z\in C$ and $t\in C\setminus \{0\}$. By taking $z=R$ and $t=\exp\left(i(\Phi-\pi/2)\right)$, it comes \begin{align} \alpha_p&=(-i)^pe^{iR\sin(\Phi-\pi/2)}\\ &=a_pe^{-iR\cos\Phi}\\ &=a_pe^{-ix} \end{align} as expected. It is however valid for all $R$.