## Composition of two formal series

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There are two formal semi-infinite Laurent series

$$f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k$$

and

$$f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}$$

Their composition $f_+(f_-(z))$ is a formal series infinite in both directions. Is there any way to construct two other semi-infinite series

$$g_+(z)=z+\sum_{k=2}^{\infty} c_k z^k$$

and

$$g_-(z)=z+\sum_{k=0}^{\infty} d_k z^{-k}$$

such that

$$f_+(f_-(z))=g_-(g_+(z))$$

How should I call this problem? Are there any non-trivial examples when it can be done?

-

It's not clear that the composition is well defined since each coefficient will be a sum of infinitely many terms. – Felipe Voloch Jun 29 at 17:13
That's true. Actually, in the "physical" example I have in my mind, coefficients are divergent. But if we assume that coefficients of the composition are well defined, what can be done? – Sasha Jun 29 at 17:23
I'm sorry.  But I am not even sure how to compute $g_-(g_+(z))$ where the $-$ part is on the outside.  For example, if $g_-(z)$ has a term $z^{-1}$ and $g_+(z)=z+z^2$, we have $$\begin{align} \left(z+z^2\right)^{-1} &= z^{-1}-1+z-z^2++z^3-z^4+\dots\quad\text{for $z$ near $0$, but} \cr \left(z+z^2\right)^{-1} &= z^{-2}-z^{-3}+z^{-4}-z^{-5}+\dots\quad\text{for $z$ near $\infty$.} \end{align}$$ – Gerald Edgar Jul 5 at 14:54
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