I am reading C. Weibel's 'The K-book', the definition of (special) $\lambda$-rings (in Page 102, Definition 4.3.1) is
A pre-$\lambda$-ring $(K,+,\cdot)$ is called a $\lambda$-ring, if the following map $\lambda_t$ is a pre-$\lambda$-ring homomorphism $$\lambda_t:K\to W(K)\quad,\quad x\mapsto \lambda_t(x)=\sum_{k=0}^\infty \lambda^k(x)t^k,$$
where the addition '$\cdot$' of the ring $W(K)=1+tK[[t]]$ is just the multiplication of polynomials (the additive identity is $1$), the multiplication '$*$' of $W(K)$ is defined by $$(1-rt)*f(t)=f(rt),$$ and the multiplicative identity is $1-t$. However the problem is $\lambda_t(1)=\sum_{k=0}^\infty \lambda^k(1)t^k$ could not equal to $1-t$, and by the classical definition of $\lambda$-rings ($\lambda^k(1)=0$ for $k\neq 0,1$), we have $\lambda_t(1)=\lambda^0(1)+\lambda^1(1)t=1+t$ is the multiplicative identity of $W(K)$. So I assume that maybe the multiplication '$*$' of ring $W(K)$ should be defined by $$(1+rt)*f(t)=f(rt)$$ and the multiplicative identity is $1+t$. I am not sure whether this is a mistake in 'The K-Book'.
Here are the pictures of the relative definition and construction in the book:
$W(K)$" />
$\lambda$-rings" />