In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator ${\mathrm{\Delta }}_{\nu }$, defined as

$${\mathrm{\Delta }}_{\nu }=\sum _{i=1}^n\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i^2}+\frac{{\nu }_i}{x_i}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},{\nu }_i\ge 0.$$

The fractional power of this operator is introduced as a pseudo-differential operator through the multi-dimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations.

We first describe commuting Toeplitz operators with harmonic symbols on weighted harmonic Bergman spaces. Then, a sufficient condition for hyponormality on weighted Bergman spaces of the punctured unit disk, when the analytic part of the symbol is a monomial, is shown.