Is there a name for series of the type $S=\sum_{k=-\infty}^{\infty}a_k(z-b_k)^k$ for some complex sequences $(a_n)_{n\in\mathbb{Z}},(b_n)_{n\in\mathbb{Z}}$ such that $S$ converges at least in some open $U\subset\mathbb{C}$? Naturally, arbitrarilly many $a_n$ may be $0$.
The closest I found are Laurent series but there $(b_n)_{n\in\mathbb{Z}}$ is constant, but I did not find anything with variable $b_n$. The finite sum $S_N=\sum_{k=-N}^{N}a_k(z-b_k)^k$ for some fixed $N\in\mathbb{N}$ would also interest me. In this case, I tried to obtain by a change of variable, using Cauchy's theorem, terms $(z-c)^k$ for some $c$ indenpendent of $k$ where all dependencies of $b_k$ are moved into $a_k$. This was highly unsuccessful.
Do those finite sums maybe have a specific name?