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!