Bose gas with generalized dispersion relation plus an energy gap

We report the critical temperature, the condensed fraction, the internal energy and the specific heat for a $d$-dimensional Bose gas with a generalized dispersion relation plus an energy gap, i.e., $\varepsilon=\varepsilon_0$ for $k=0$ and $\varepsilon=\varepsilon_0 +\Delta+ c_sk^s$, for $k>0$, where $\hbar k$ is the particle momentum, $\varepsilon_0$ the lowest particle energy, $c_s$ a constant with dimension of energy multiplied by a length to the power $s > 0$. When $\Delta > 0$, a Bose-Einstein critical temperature $T_c \neq 0$ exists for any $d/s \geq 0$ at which the internal energy shows a peak and the specific heat shows a jump. The critical temperature and the specific heat jump increase as functions of the gap but they decrease as functions of $d/s$. Thermodynamic properties are $\varepsilon_0$ independent since this is just a reference energy. For $\Delta = 0$ we recover the results reported in Ref. [1].\\ [1] V. C. Aguilera-Navarro, M. de Llano y M. A. Sol\'is, Eur. J. Phys. {\bf 20}, 177 (1999).