What is the correct name for a "product function" on a monoid?

Question

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-commutative arithmetic sequence". I'd like to use the correct name to refer to these functions as well as the related monoid of "formal product functions in an unknown variable", isomorphic to $W * \langle x \rangle$, the free product of W and the free monoid in a single generator, $\langle x\rangle = \{1,x,x^2,x^3,\ldots\}$. These have the same relation as polynomial functions in a ring $R$ to formal polynomials with coefficients in $R$.