Equivalence of ideal gases in two dimensions and Landen's relations

Based on a recently given unified formulation of the statistical thermodynamics of ideal gases, an equivalence between the Bose and Fermi gases in two dimensions is established. This equivalence is shown to be a consequence of Landen's relations, found some 200 years ago, relating the Bose and Fermi fugacities through the Euler transformation. The specific heat result due to May [Phys. Rev. 135, A1515 (1964)] follows at once. The special nature of two-dimensional solutions is described.