  Recent work in the literature has studied a version of non-commutative
Schwarzschild black holes where the effects of non-commutativity are described
by a mass function depending on both the radial variable r and a
non-commutativity parameter theta. The present paper studies the asymptotic
behaviour of solutions of the zero-rest-mass scalar wave equation in such a
modified Schwarzschild space-time in a neighbourhood of spatial infinity. The
analysis is eventually reduced to finding solutions of an inhomogeneous
Euler--Poisson--Darboux equation, where the parameter theta affects explicitly
the functional form of the source term. Interestingly, for finite values of
theta, there is full qualitative agreement with general relativity: the
conformal singularity at spacelike infinity reduces in a considerable way the
differentiability class of scalar fields at future null infinity. In the
physical space-time, this means that the scalar field has an asymptotic
behaviour with a fall-off going on rather more slowly than in flat space-time.
