AUTOMORPHISMS OF THE PLANAR TREE POWER SERIES ALGEBRA AND THE NON-ASSOCIATIVE LOGARITHM
Lothar Gerritzen · Bulgarian Digital Mathematics Library (BulDML) at IMI-BAS (Institute of Mathematics and Informatics) · 2004
In this note we present the formula for the coefficients of the substitution series f (g(x)) of planar tree power series g(x) into f (x).The coefficient c T (f (g(x))) relative to a finite planar reduced rooted tree T is the sumwhich is extended over all open subtrees S of T .In this expression c S (f ) is the coefficient of f with respect to S and T -In(S) is the complement of the inner vertices of S in T while c (T -In(S)) (g) is the product of the coefficients of g relative to the connected components of the forest T -In(S).Also we give the formula for the compositional inverse of a planar tree power series g in terms of coefficients of g.It is shown that the generic non-associative exponential series EXP has an inverse LOG.Results about coefficients of LOG are presented.