# 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!

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