Abstract
In this paper, we investigate the fractional power (-Delta(nu))(gamma/2), 0 < gamma < 2, of the Bessel operator in the form Delta(nu) := d(2)/dx(2) + 2 nu+1/x d/dx, nu > -1/2. Our method uses a canonical representation for generalized Laplacian related to the structures of the Bessel- Kingman hypergroup. As a direct application, we solve the space-fractional diffusion equation.