  A unified description for the Bose and Fermi gases trapped in an external
generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ 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.
