Is there a shortcut to summing Bessel functions with imaginary units?

[matematikuvol]

Homework Statement

What is easiest way to summate
$$\sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}]$$
where $i$ is imaginary unit.

Homework Equations

The Attempt at a Solution

I don't need to write explicit Bessel function so in sum could stay
$$C_1J_(x)+C_2J_2(x)+...$$
Well I see that terms in the sum will be
$$2iJ_1(x)-2J_2(x)+...$$
But I search for more sofisticated solution. Is there any way to sum this using $i^1=i$,$i^2=-1$,$i^3=-i$,$i^4=1$?

 

[lurflurf]

cute

first i^n+(-1)^n i^-n=2 i^n

then split into sums over

2k
2k+1

Which have nice well know sums involving sin(x) and J0(x)