polynomials - Confusing Sequence-Notation

I recently encountered the following definition:

• there exists a sequence of polynomials $(p_n)_{n\in\mathbb{N}}$ of fixed
  degree $\sigma$ such that for every $x\in U$,
  $\left|f(n+x) - p_n(n+x) \right| \longrightarrow 0
  \quad \text{as} \;n\to+\infty$

What would one of these polynomials look like?

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I have seen polynomial sequences of the form $p_n(n^2)$, which would expand as $1, 4, 9...$ as well as polynomials of the form $p_n(x) = \sum_{k=0}^n a_nx^k$
In these cases $n$ was always the degree of the polynomial, though here we have polynomials of a fixed degree as well as polynomials in terms of $n+x$ as opposed to just $x$

I would assume that the polynomials in question would be in one of the following forms:

$$\begin{align}

&1)\qquad p_n(n+x) = \sum_{k=0}^{\sigma}a_{n+x}(n+x)^k\\
&2)\qquad p_n(n+x) = \sum_{k=0}^{\sigma}a_{n+x}(n)^k\\
&3)\qquad p_n(n+x) = \sum_{k=0}^{\sigma}a_{n}(n+x)^k
\end{align}
$$

However, I am not quite sure which (if any) of these forms I should be looking at, which is hurting my understanding of the topic in general

The paper from which this comes claims that the natural logarithm can be approximated in this fashion by a sequence of degree $0$ using the above definition, and I am trying to find such a sequence. IF anyone could provide an explicit example that would be great as well!

As noted in the comments, the paper in question (as well as additional constraints on $f$ which maps $\mathbb{C} \to \mathbb{C}$) can be found in the linked question, though I do include such constraints here as I find them unnecessary to my specific question