Understand variant kinds of generating functions?

I found there are various kinds of generating functions in Wikipedia. I would like to understand why (the purpose)and how these concepts were created.

For the "how" part, given a sequence $(a_n)$,

• the ordinary generating function is defined as $(a_n)$-weighted version of the Taylor expansion of $(1-x)^{-1}$ at $x=0$;

• the exponential generating function is defined as $(a_n)$-weighted version of the Taylor expansion of $e^x$ at $x=0$.

I was wondering if the following two kinds can be viewed as $(a_n)$-weighted versions of the Taylor expansions of some functions at some points:

• The Poisson generating function of a sequence $(a_n)$ is $$\operatorname{PG}(a_n;x)=\sum _{n=0}^{\infty} a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x)\,.$$ If ignoring $a_n$, $\operatorname{PG}(a_n;x)$ seems to expand $1$ by writing it as $1=e^{-x} e^x$ and expand the second factor by the exponential generating function.

• The Lambert series of a sequence $(a_n)$ is $$\operatorname{LG}(a_n;x)=\sum _{n=1}^{\infty} a_n \frac{x^n}{1-x^n}.$$

Are the following two kinds viewed as weighted versions of some kinds of expansions of some functions at some points?

• The Bell series of a sequence $(a_n)$ is an expression in terms of both an indeterminate x and a prime p and is given by $$\operatorname{BG}_p(a_n;x)=\sum_{n=0}^\infty a_{p^n}x^n.$$
• Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence an is $$\operatorname{DG}(a_n;s)=\sum _{n=1}^{\infty} \frac{a_n}{n^s}.$$

Thanks and regards!

You could consider them as sort-of Fourier expansions, just that you only ask for the base functions ($z^n$ for ordinary, $z^n/n!$ for exponential, $\mathrm{e}^{-z} z^n/n!$ for Poisson, $1/n^z$ for Dirichlet) to be a linearly independent set, no orthogonality required; i.e, as a vector space where you are interested in the representation of a vector on the given base.
The point of the different bases (and the power of generating functions) lies in that operations on the series represent interesting operations on the sequence on display. Adding generating functions is clearly adding corresponding terms, multiplication gives for example: \begin{align} \left(\sum_{n \ge 0} a_n z^n\right) \cdot \left(\sum_{n \ge 0} b_n z^n\right) &= \sum_{n \ge 0} \left(\sum_{0 \le k \le n} a_k b_{n - k} \right) z^n \\ \left( \sum_{n \ge 0} a_n z^n / n! \right) \cdot \left( \sum_{n \ge 0} b_n z^n / n! \right) &= \sum_{n \ge 0} \left(\sum_{0 \le k \le n} \binom{n}{k} a_k b_{n - k} \right) z^n / n! \end{align} One can interpret the convolutions above as "make up a new object of size $m + n$ out of all combinations of an $m$-sized $A$ with an $n$-sized $B$", for the cases of unlabeled (not ordered) pieces for ordinary generating functions, and labeled (ordered) pieces for exponential generating functions. The others are much less used, but in their cases multiplication has also an interesting interpretation. The point being that it is often much easier to manipulate the generating function as one object instead of the separate coefficients.