I am reading C. Weibel's 'The K-book', and in page 101, example 4.3, there is a construction of the ring $(W(K)=1+tK[[t]],\cdot,*)$. I could not understand how can we construct the multiplication '$*$' according to the method in this book. Here is the picture of this example:
According to the formal factorization of elements $$f(t)=\Pi_{i=1}^\infty(1-r_it^i),$$ to get 'universal polynomials' $P_n$, I think it is sufficient to just consider the multiplication $(1-r_1t^m)*(1-r_2t^n)$ for any $r_1,r_2\in K$ and $m,n\geq 1$. However I do not know how to get this just by the equation $$(1-rt)*f(t)=f(rt).$$
The question seems to need only the functor $W$ on fields, so only the object $W(K)$ for a field $K$ was addressed, for it only the multiplication $*$. By naturality / functoriality, consider the algebraic closure $\bar K$ of $K$, then computations in $W(K)$ are "the same" as in $W(\bar K)$.
In this last Witt ring, consider first - as in the question - the $*$-product of only two elements, as in the question: $$ \begin{aligned} f &= f(t) = 1-rt^m\ ,\\ g &= g(t) = 1-st^n\ . \\ &\qquad\text{ Over $\bar K$ we can split in linear factors, say:} \\ 1-rU^m &=(1-r_1U)(1-r_2U)\dots(1-r_m U)\ ,\\ 1-sV^n &=(1-s_1V)(1-s_2V)\dots(1-s_n V)\ ,\\ &\qquad\text{ so that using the given receipt of star-multiplication with $1-at$ elements:} \\ f*g &=(1-rt^m)*(1-st^n)\\ &= \Bigg(\ (1-r_1t)(1-r_2t)\dots(1-r_m t)\ \Bigg) * \Bigg(\ (1-s_1t)(1-s_2t)\dots(1-s_n t)\ \Bigg) \\ &=\prod_{j,k}(1-r_j t)*(1-s_kt)\qquad\text{(distributivity)} \\ &:=\prod_{j,k}(1-r_js_k\;t)\ . \\ &\qquad\text{ Or also:} \\ f*g &=(1-rt^m)*(1-st^n)\\ &= \Bigg(\ (1-r_1t)(1-r_2t)\dots(1-r_m t)\ \Bigg) * (1-st^n) \\ &=\prod_j(1-s\; r_j^n\; t^n)\qquad\text{(distributivity)}\ . \\ &\qquad\text{ Or also:} \\ f*g &=\prod_k(1-r\; s_k^m\; t^m)\ . \end{aligned} $$ It is clear that in either representation we have a polynomial $f*g$ in $t$ with coefficients that are symmetric polynomials in the $r$-tuple of variables $(r_1,r_2,\dots,r_m)$ on the one side, and in the $s$-tuple $(s_1,s_2,\dots,s_n)$ on the other side. So these coefficients are polynomials in the elementary polynomials for these tuples, which are $0,0,\dots,0,r$ for the $r$-tuple, and $0,0,\dots,0,s$ for the $s$-tuple. So we have as result a polynomial of degree $mn$ with coefficients (universal) polynomials in $r,s$.
We have then a definition of the product of truncated formal factorizations $$ \left(\prod_m(1-r_mt^m) \quad\text{ modulo }t^N\right)\quad *\quad \left(\prod_n(1-s_n t^n)\quad\text{ modulo }t^N\right) $$ and let finaly $N$ go to infinity. This is ok, the operations are compatible with the $t^N$ piece filtration, since the star product with a "term" $(1-rT^N)$ introduces by definition only further "terms" of the shape $(1-r\;s_k^N\;t^N)$, as noted above.