  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 $q\in\mathbb C^\times=\mathbb C\setminus\{ 0\}$ we
construct Fr\'echet 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. We show that $\mathcal O_q(\mathbb D^n)$ and
$\mathcal O_q(\mathbb B^n)$ are not isomorphic provided that $|q|=1$ and $n\ge
2$. This result can be interpreted as a $q$-analog of Poincar\'e's theorem,
which asserts that $\mathbb D^n$ and $\mathbb B^n$ are not biholomorphically
equivalent unless $n=1$. In contrast, $\mathcal O_q(\mathbb D^n)$ and $\mathcal
O_q(\mathbb B^n)$ are shown to be isomorphic for $|q|\ne 1$. Next we prove that
$\mathcal O_q(\mathbb D^n)$ is isomorphic to a quotient of J. L. Taylor's "free
polydisk algebra" (1972), while $\mathcal O_q(\mathbb B^n)$ is isomorphic to a
quotient of G. Popescu's "free ball algebra" (2006). This enables us to show
that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ yield strict
deformation quantizations (in Rieffel's sense) of $\mathcal O(\mathbb D^n)$ and
$\mathcal O(\mathbb B^n)$, respectively. Finally, we study relations between
our deformations and formal deformations of $\mathcal O(\mathbb D^n)$ and
$\mathcal O(\mathbb B^n)$.
  Some of the results of this paper were announced in arXiv:1311.0309
[math.FA].
