Algebraic complexity theory studies the number of arithmetic operations necessary to compute different polynomials. As in the case of Boolean complexity, there are many open questions: indeed, there are several natural polynomials for which we believe, but are unable to prove, that no efficient computation is possible. When contrasted with the difficulty to obtain lower bounds in general for the main computation model of arithmetic circuits, Nisan’s 1991 result is striking: it gives an exact characterization of the complexity of computing any given homogeneous polynomial with a homogeneous algebraic branching program, in the non-commutative setting. As noticed in 2020 by Fijalkow et al., Nisan’s theorem can be derived from results by Fliess (1974) in the field of non-commutative formal series. We build on this point of view to extend Nisan’s theorem to general (not necessarily homogeneous) polynomials computed by general algebraic branching programs and again obtain an exact characterization. Fijalkow et al. showed how to apply properties of formal tree series to get a similar statement for homogeneous non-associative non-commutative arithmetic circuits and we also extend this characterization to general non-associative non-commutative circuits.