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Past research has shown that application of mathematical and geometrical concepts such as fractals, aperiodic tilings, and special polynomials can provide elegant solutions to difficult antenna array design problems. For example, design issues such as beam shaping and control, sidelobe levels, bandwidth and many others have been addressed with such concepts. In this paper, mathematical constructs based on the raised power series (RPS) are utilized to provide easily controlled aperiodicity to a linear array of antenna elements in order to achieve wideband performance. In addition, recursive application of raised power series subarrays and implementation of an optimization technique based on the genetic algorithm is demonstrated to realize impressive ultrawideband performance. The technique introduced here is shown to offer bandwidths of many octaves with excellent sidelobe suppression and no grating lobes. Moreover, the ultrawideband performance for one of the optimized RPS array examples is verified through full-wave simulations which take into account the coupling environment experienced by realistic radiating elements (in this case half-wave dipole antennas for three different operating frequencies).