#### The Chen-Fliess series

The P. Hall basis from Section 15.4.3 applies in general to any Lie algebra. Let , , denote a P. Hall basis for the nilpotent formal Lie algebra . An important theorem in the study of formal Lie groups is that every can be expressed in terms of the P. Hall basis of its formal Lie algebra as

 (15.127)

which is called the Chen-Fliess series. The are sometimes called the backward P. Hall coordinates of (there is a forward version, for which the terms in (15.127) go from to , instead of to ).

Steven M LaValle 2020-08-14