Roots in the substitution group and in the group of Riordan matrices with ones in the main diagonal

Abstract

We investigate the existence and uniqueness of iterative roots of order $n$ within the substitution group of formal power series $\mathcal{J}(Z)$ -- with coefficients in a commutative ring with unity $Z$ -- employing a matrix-based framework grounded in the Riordan group. We analyse the relationship between the substitution group and the Lagrange subgroup -- a group of Riordan matrices -- and explore some classic questions concerning algebraic completeness and uniqueness of root extractions. This approach allows us to obtain various results about the roots in $\mathcal{J}(Z)$ for different choices of $Z$. Furthermore, the examination of the substitution group facilitates the analysis of roots within the Riordan matrix group.