Let's treat $z^{1/T}$ and $w^{1/T}$ as multivalued complex functions. Then, what does $(z+w)^{1/T}$ exactly mean?
We have two ways to interprate $(z+w)^{1/T}$: $$I=\sum_{i}\binom{1/T}{i}z^{1/T}z^{-i}w^i$$ and $$II=\sum_{i}\binom{1/T}{i}w^{1/T}w^{-i}z^i$$.
My question: when viewed as multivalued functions, are the above two series the expansions of the same (multivalued) function on different domains?
Moreover, if we interpret multivalued functions $z^{1/T}$ and $w^{1/T}$ as holomorphic functions on the universal covering of $(\mathbb{C}^2,0)$, how to interpret $(z+w)^{1/T}$?