  We prove existence and uniqueness of a solution of the Dirichlet problem for
separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb
D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson
kernel. A characterization by hypergeometric functions of such functions which
are also m - homogeneous is given, this characterization is used to obtain
series expansion of these functions. Basic $H^p$ theory of such functions is
developed: integral representations by measures and $L^p$ functions on $\mathbb
T^n$, norm and weak star convergence at the distinguished boundary $\mathbb
T^n$. Weak $(1, 1)$ - type estimate for a restricted non-tangential maximal
function is derived. Slice functions $u(z_1, . . . , z_k, \zeta_{k+1}, . . . ,
\zeta_n)$, where some of the variables are fixed, are shown to belong in the
appropriate space of functions of $k$ variables. We prove a Fatou type theorem
on a. e. existence of restricted non-tangential limits for these functions and
a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our
results extend earlier results for $(\alpha, \beta)$ harmonic functions in the
disc and for n - harmonic functions in $\mathbb D^n$.

Complex Variables