Unified statistical thermodynamics of quantum gases trapped under generic power law potential in $d$ dimension and equivalence in $d=1$

A unified description for the Bose and Fermi gases trapped in an external generic power law potential $U={\sum }_{i=1}^d{c}_i\mathrm{\mid }\frac{x_i}{a_i}{\mathrm{\mid }}^{n_i}$ is presented using the grandpotential of the system in $d$ dimensional space. The thermodynamic quantities of the quantum gases are derived from the grand potential. An equivalence between the trapped Bose and Fermi gases is constructed in one dimension ($d=1$) using the Landen relation. It is also found that the established equivalence between the ideal free Bose and Fermi gases in $d=2$ (M. H. Lee, Phys. Rev. E 55, 1518 (1997)) is lost when external potential is applied.