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Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = {pn: n~>0}. We characterize and analyze those bases from which one can construct a k-algebra of 'formal B-series' of the form f= ~c,p,, with cn Ek, showing inter alia that many classical polynomial bases fail to have this property.

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