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?
