## Algebraic Formulae for the $q$-inverse in a Free Group

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 4, Page 721--734.

Habiro, K. ; Kamada, S. ; Matsumoto, Y. ; Yoshikawa, K.
Algebraic Formulae for the $q$-inverse in a Free Group
Let $F_n$ be a free group with $n$ fixed generators, which we assume are linearly ordered. In a previous paper [\rM], a curious mapping $I: F_n \to F_n$ was introduced pictorially. It is a \lq\lq square root\rq\rq\ of the inner automorphism of $F_n$ induced by the \lq\lq smallest\rq\rq\ generator. In the present paper, two algebraic formulae will be given, by which one can compute the mapping $I$ purely algebraically.