ContributionsFinitary bases and formal generating functions

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 = p_n: 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_n p_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.