A power-commutator presentation for the group of truncated polynomials under substitution

Given a prime $p$, the Nottingham group $G(p)$ is the group of formal power series $x+a_2{x}^2+a_3{x}^3+\cdots$, $a_i\in \mathbf{Z}\mathrm{/}p\mathbf{Z}$, under substitution. For $n\ge 1$, we write $G_n(p)$ for the quotient of $G(p)$ by its normal subgroup $K_n=\{x+a_{n+1}{x}^{n+1}+a_{n+2}{x}^{n+2}+\cdots \mathrm{\mid }{a}_i\in \mathbf{Z}\mathrm{/}p\mathbf{Z}\}$, namely the group of truncated polynomials $x+a_2{x}^2+\cdots +a_n{x}^n$, $a_i\in \mathbf{Z}\mathrm{/}p\mathbf{Z}$, under substitution, which is a $p$-group of order $p^{n-1}$ generated by $x+x^2,\dots ,x+x^n$. In this paper, we determine the power-commutator presentation of $G_{15}(p)$ relative to $x+x^2,\dots ,x+x^{15}$. This, in turn, automatically gives the power-commutator presentation of $G_n(p)$ relative to $x+x^2,\dots ,x+x^n$ for any $1\le n<15$, as well as the initial segments of the power and commutator relations in $G_n(p)$ corresponding to $x+x^2,\dots ,x+x^{15}$ for any $n>15$.