Sampling zeros and the Euler-Frobenius polynomials

We show that the zeros of sampled-data systems resulting from rapid sampling of continuous-time systems preceded by a zero-order hold (ZOH) are the roots of the Euler-Frobenius polynomials. Using known properties of these polynomials, we prove two conjectures of Hagiwara et al. (1993), the first of which concerns the simplicity, negative realness, and interlacing properties of the sampling zeros of ZOH- and first-order hold (FOH-) sampled systems. To prove the second conjecture, we show that in the fast sampling limit, and as the continuous-time relative degree increases, the largest sampling zero for FOH-sampled systems approaches 1/e, where e is the base of the natural logarithm.