so I'm currently struggling witht the following problem:
Show that the maps $$e_k: \mathbb{N}_0 \rightarrow \mathbb{Z}; x \mapsto \frac{x(x-1) \cdots (x-k+1)}{k!}$$ for $k \in \mathbb{N}_0$ (with $e_0$ identically $1$) form a basis for Int$(\mathbb{Z})$ as a $\mathbb{Z}$-module.
In case that this is not a widespread notation, Int$(\mathbb{Z})$ refers to the functions that map integers to integers.
To be quite honest, I am quite confused by the question, I don't understand the maps. I don't understand why one has a normal arrow and one has an arrow with a little base. what are the maps supposed to mean? Any help would be really appreciated!
The MSE question 4049277 "Avoiding brute force: determining when a specific polynomial in $\mathbb{Q}[x]$ is an integer for any integer $x$" asks essentially the same question. Some of the answers are for the general case. My answer used the technique of a difference table to solve the general problem of expressing an integer valued polynomial as a $\mathbb{Z}$-linear combination of binomial coefficients. The formula is $$ f(x) = \sum_{k=0}^\infty \Delta^k\!f(0) {x \choose k} $$ where the sum has only a finite number of non-zero terms since $\,f(x)\,$ is a polynomial function.