I am trying to understand what generating functions are, so I looked at Wikipedia.
Excerpt:
In mathematics, a generating function is a way of encoding an infinite sequence of numbers $(a_n)$ by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence.
...
The ordinary generating function of a sequence $a_n$ is
$$G(a_n;x) = \sum_{n = 0}^{\infty} a_n x^n.$$
Discussion:
I am trying to understand this definition relative to things I already have some familiarity with. I know a small amount about formal power series and formal power series rings.
I suppose that if we have a sequence $(a_n)$ and ring $R$ such that all $a_i \in R$, then a generating function $g$ is just an element of the formal power series ring $R[[X]]$. However, presumably there are generating functions that do not belong to a ring of formal power series, because the components of the sequence are not required to be elements of a ring.
On second thought, I'm not sure it makes sense to consider sequences $(a_n)$ whose components are not ring elements, because then as far as I know your formal power series is just a string $a_0 + a_1 X + a_2 X^2 + \dots$ without any additional structure, in which case the definition of a generating function does not lead to any gain, since you're still just dealing with a sequence.
So, would it be appropriate to think of a generating function as an element of a ring of formal power series?
Thanks for your help.
