Further reductions of Poincaré-Dulac normal forms in ${\mathbf{C}}^{n+1}$

In this paper, we will consider (germs of) holomorphic mappings of the form $(f(z),\lambda _{1} w_{1}(1+g_{1}(z)),\ldots ,\lambda_{n}w_{n}(1+g_{n}(z)))$, defined in a neighborhood of the origin in ${\mathbf{C}}^{n+1}$. Most of our interest is in those mappings where $f(z)=z+a_{m}z^{m}+\cdots$ is a germ tangent to the identity and $g_{i}(0)=0$ for $i=1,\ldots ,n$, and $\lambda _{i}\in {\mathbf{C}}$ possess no resonances, for these are the so-called Poincaré-Dulac normal forms of the mappings $(z+O(2), \lambda _{1}w+O(2),\ldots ,\lambda _{n}w+O(2))$. We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.