Holomorphic functions on the quantum polydisk and on the quantum ball

We introduce and study noncommutative (or ``quantized'') versions of the algebras of holomorphic functions on the polydisk and on the ball in \(\mathbb C^n\). Specifically, for each \(q\in\mathbb C\setminus\{ 0\}\) we construct Fréchet algebras \(\mathcal O_q(\mathbb D^n)\) and \(\mathcal O_q(\mathbb B^n)\) such that for \(q=1\) they are isomorphic to the algebras of holomorphic functions on the open polydisk \(\mathbb D^n\) and on the open ball \(\mathbb B^n\), respectively. In the case where \(0