$k$. Every element of $R[X][[Y]]$ can be written uniquely in the form $$\begin{aligned}
 f &=& \sum_{k=0}^\infty f_{k}(X, Y),\end{aligned}$$ where $f_k(X, Y)$ is an element of $S_k$. In this expansion, there is no restriction on $\deg_X f_k(X, Y)$. Motivated by several applications to the $p$-adic theory of zeta functions over finite fields, we want to consider subrings of $R[X][ Y]]$ in which $\deg_X (f_k)$ is bounded above by some function $\lambda$. In particular, let $\lambda : {\mathbf{R}}_{\geq 0} \rightarrow {\mathbf{R}}_{\geq 0}$ be a monotonic up function. We call $\lambda$ a growth function. Following Wan [@Wan1], we define a subring of $R[X][[Y]]$ as follows: $$\begin{aligned}
 R[X; Y, \lambda] &=& \{f = \sum_{k=0}^\infty f_{k}(X, Y) :
 f_k  \in S_k, \deg_X(f_k) \leq C_f\lambda(k), \text{ for } k \gg 0 \},\end{aligned}$$ where $C_f$ is a constant depending only on $f$. Since $\lambda$ is monotonic up, it satisfies the trivial inequality, $$\begin{aligned}
 \lambda(x) + \lambda(y) &\leq& 2\lambda(x+y)\end{aligned}$$ for all $x$ and $y$ in ${\mathbf{R}}_{\geq 0}$. From this inequality, it is clear that $R[X; Y, \lambda]$ is an $R[X]$-algebra, which contains $R[X]$.

If $\lambda$ is invertible, we have the following equivalent definition: $$\begin{aligned}
 R[X; Y, \lambda] &=& \{g = \sum_{d=0}^\infty g_{d}(X, Y) : g_d  \in A_d, {\text{ord}}_Y(g_d) 