  We show that the space $\mathcal{H}(\Omega)$ of holomorphic functions
$F:\Omega\to\mathbb{C}$, where ${\Omega=\{(z,w)\in\widehat{\mathbb{C}}^2\,:\,
z\cdot w\neq 1\}}$, possesses an orthogonal Schauder basis consisting of
distinguished eigenfunctions of the canonical Laplacian on $\Omega$. Mapping
$\Omega$ biholomorphically onto the complex two-sphere, we use the Schauder
basis result in order to identify the classical three-dimensional spherical
harmonics as restrictions of the elements in $\mathcal{H}(\Omega)$ to the real
two-sphere analogue in $\Omega$. In particular, we show that the zonal
harmonics correspond to those functions in $\mathcal{H}(\Omega)$ that are
invariant under automorphisms of $\Omega$ induced by M\"obius transformations.
The proof of the Schauder basis result is based on a curious combinatorial
identity which we prove with the help of generalized hypergeometric functions.
