  In this paper, we study free holomorphic functions on regular polyballs and
provide analogues of several classical results from complex analysis such as:
Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum
principle. These results are used together with a class of noncommutative
Berezin transforms to obtain a complete description of the group Aut(B_n) of
all free holomorphic automorphisms of the polyball. The abstract polyball B_n
has a universal model S consisting of left creation operators acting on tensor
products of full Fock spaces. We determine: the group of automorphisms of the
Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball
algebra A_n; the group of unitarily implemented automorphisms of the polyball
algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We
prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact,
locally compact topological group with respect to the topology induced by an
appropriate metric. Finally, we obtain a concrete unitary projective
representation of the topological group Aut(B_n)in terms of noncommutative
Berezin kernels associated with regular polyballs.
