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We analyze micro-ring resonator chains that have a system response with N-th order poles and zeroes when the chain consists of N unit cells. These include the recently proposed zigzag micro-ring resonator chains exhibiting new fast- and slow- light properties and a dual ring chain for which the explicit expression of the system function is given. We discuss the realization of ultrahigh order structures with sharp peaks and large out-of-band rejection filters, ultra-sharp notch optical filters for optical carrier suppression and the general properties of slow and fast light phenomena in this type of systems. All the analyses are based on the pole-zero representation of the system function. A tuning mechanism for the modified dual-ring structures acting as notch filters is also proposed.