I am reading about the Siegel Center Problem and I came across a paper that uses notation I am not familiar with. They say,
In this paper, we study the Siegel center problem. Consider two subalgebras $A_1 \subset A_2$ of $z\mathbb{C}[[z]]$ closed with respect to the composition of formal series. For example, $z\mathbb{C}[[z]]$, $z\mathbb{C}\{z\}$ (the usual analytic case), ...
I am specifically studying the center problem in the analytic case i.e. for holomorphic functions, $$f(z) = \lambda z + a_2z^2 + a_3z^3 + ...$$ which are defined throughout some neighbourhood of the origin, with a fixed point of multiplier $\lambda$ at the origin.
Source: http://www.numdam.org/article/BSMF_2000__128_1_69_0.pdf (free access)
The notation $\mathbb{C}[[z]]$ represents the set of formal power series with complex coefficient in the variable $z$, endowed with the "obvious" ring structure. That is, $$ \mathbb{C}[[z]] := \left\{ \sum_{n=0}^{\infty} \alpha_n z^n : \alpha_n \in \mathbb{C} \right\}, $$ where addition and multiplication are defined in the way that you would expect from finite series, i.e. $$ \sum \alpha_n z^n + \sum \beta_n z^n = \sum (\alpha_n + \beta_n) z^n $$ and $$ \left( \sum_{n=0}^{\infty} \alpha_n z^n \right) \left( \sum_{n=0}^{\infty} \beta_n z^n \right) = \sum_{n=0}^{\infty} \sum_{k=0}^{n} \alpha_k \beta_{n-k} z^n. $$
Informally, the notation $z \mathbb{C}[[z]]$ denotes the subset of formal power series that we get by multiplying every element of $\mathbb{C}[[z]]$ by $z$. With slightly more rigor, \begin{align} z\mathbb{C}[[z]] &:= \left\{ z \cdot S : S \in \mathbb{C}[[z]] \right\} \\ &= \left\{ z\sum_{n=0}^{\infty} \alpha_n z^n : \alpha_n \in \mathbb{C} \right\} \\ &= \left\{ \sum_{n=1}^{\infty} \alpha_n z^{n} : \alpha_n \in \mathbb{C} \right\}. \end{align} Note that this corresponds to the set of formal power series without a constant term.