# Fourier series transformation help - unable to reconstruct (simple) paper results

I am reading a paper and am having difficult reconstructing an equation. The paper begins with the following equations where $$\omega_{n} = \omega_{o} + n\Delta\omega$$ and:

(1) $$A_{in}(t) = \sum_{n}a_{n}(t)e^{-i\omega_{n}t}$$

(2) $$B_{out}(t) = \sum_{n}b_{n}(t)e^{-i\omega_{n}t}$$

(3) $$B_{out}(t) = e^{i\phi(t)}A_{in}(t)$$

From the above equations ($$\phi(t)$$ is $$\frac{2\pi}{\Delta\omega}$$ periodic), and using Fourier series coefficients of $$e^{i\phi(t)}$$, they state:

(4) $$b_{m}(t) = \sum_{n=-\infty}^{\infty}c_{m-n}a_{n}(t)$$

(5) $$c_{k} = \frac{\Delta\omega}{2\pi}\int_{0}^{\frac{\Delta\omega}{2\pi}}dte^{i\phi(t)}e^{ik\Delta\omega t}$$

I have been trying to confirm equations (4) and (5) without success. Specifically, I am not sure how they reduce the sum on the left side to the mth element. I appreciate any and all help! Thanks