This paper provides a brief history of B-series and the associated Butcher
group and presents the new theory of word series and extended word series.
B-series (Hairer and Wanner 1976) are formal series of functions parameterized
by rooted trees. They greatly simplify the study of Runge-Kutta schemes and
other numerical integrators. We examine the problems that led to the
introduction of B-series and survey a number of more recent developments,
including applications outside numerical mathematics. Word series (series of
functions parameterized by words from an alphabet) provide in some cases a very
convenient alternative to B-series. Associated with word series is a group G of
coefficients with a composition rule simpler than the corresponding rule in the
Butcher group. From a more mathematical point of view, integrators, like
Runge-Kutta schemes, that are affine equivariant are represented by elements of
the Butcher group, integrators that are equivariant with respect to arbitrary
changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552
