Cartan-Thullen theorem for $\mathbb C^n$-holomorphic convexity and a related problem

Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=j$ implies $\frac{\partial f_i}{\partial z_j}=0$, where $(z_1,z_2,\cdots,z_n) \in \mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$ holds. We note that a $\mathbb C^n$-analytic mapping and a $\mathbb C^n$-analytic manifold can also be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for $\mathbb C^n$-analytic functions. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $\mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, \cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following very partial result. If an open $\mathbb C^n$-holomorphically convex set $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $\mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i