  We consider a class of weighted harmonic functions in the open upper
half-plane known as $\alpha$-harmonic functions. Of particular interest is the
uniqueness problem for such functions subject to a vanishing Dirichlet boundary
value on the real line and an appropriate vanishing condition at infinity. We
find that the non-classical case ($\alpha\neq0$) allows for a considerably more
relaxed vanishing condition at infinity compared to the classical case
($\alpha=0$) of usual harmonic functions in the upper half-plane. The reason
behind this dichotomy is different geometry of zero sets of certain polynomials
naturally derived from the classical binomial series.
  Our findings shed new light on the theory of harmonic functions, for which we
provide uniqueness results under vanishing conditions at infinity along a)
geodesics, and b) rays emanating from the origin. The geodesic uniqueness
results require vanishing on two distinct geodesics which is best possible. The
ray uniqueness results involves an arithmetic condition which we analyze by
introducing the concept of an admissible function of angles. We show that the
arithmetic condition is to the point and that the set of admissible functions
of angles is minimal with respect to a natural partial order.
