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When Ishiin was fighting arrankar Grandfisher, Grandfisher said "that's why Ichigo's so strong, he's a pure blood" implying that both Ishiin and Ichigo's mum were shinigami that may be the reason of his accelerated growth in learning advanced shinigami techniques.
As for his hollow powers i think that's got something to do with all his exposure to hollows and shinigami before dying and becoming one himself, and also the shaft, the hougyoku though i doubt it is genetic.
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