  It is known that at least ten equivalent definitions of the fractional
Laplacian exist in an unbounded domain. Here we derive a further equivalent
definition that is based on the Mellin transform and it can be used when the
fractional Laplacian is applied to radial functions. The main finding is tested
in the case of the space-fractional diffusion equation. The one-dimensional
case is also considered, such that the Mellin transform of the Riesz (namely
the symmetric Riesz--Feller) fractional derivative is established. This
one-dimensional result corrects an existing formula in literature. Further
results for the Riesz fractional derivative are obtained when it is applied to
symmetric functions, in particular its relation with the Caputo and the
Riemann--Liouville fractional derivatives.
