Moment Series and R-transform of the Generating Operator of $L(F_N)$

In this paper, we will consider the free probability theory of free group factor $L(F_N),$ where $F_k$ is the free group with $k$-generators. We compute the moment series and the R-transform of the generating operator, $T=g_1+\mathrm{.}\mathrm{.}\mathrm{.}+g_N+g_1^{-1}+\mathrm{.}\mathrm{.}+g_N^{-1}\mathrm{.}$