Mellin definition of the fractional Laplacian

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