  Bose-Einstein condensation in a Bose gas is studied analytically, in any
positive dimensionality ($d>0$) for identical bosons with any energy-momentum
positive-exponent ($s>0$) plus an energy gap $\Delta$ between the ground state
energy $\varepsilon_0$ and the first excited state, i.e.,
$\varepsilon=\varepsilon_0$ for $k=0$ and $\varepsilon=\varepsilon_0 +\Delta+
c_sk^s$, for $k>0$, where $\hbar \mathbf{k}$ is the particle momentum and $c_s$
a constant with dimensions of energy multiplied by a length to the power $s >
0$. Explicit formula with arbitrary $d/s$ and $\Delta$ are obtained and
discussed for the critical temperature and the condensed fraction, as well as
for the equation of state from where we deduce a generalized $\Delta$
independent thermal de Broglie wavelength. Also the internal energy is
calculated from where we obtain the isochoric specific heat and its jump at
$T_c$. When $\Delta > 0$, a Bose-Einstein critical temperature $T_c \neq 0$
exists for any $d > 0$ at which the internal energy shows a peak and the
specific heat shows a jump. Both the critical temperature and the specific heat
jump increase as functions of the gap but they decrease as of $d/s$. At
sufficiently high temperatures $\Delta$- independent classical results are
recovered. However, for temperatures below the critical one the gap effects are
predominant. For $\Delta = 0$ we recover previous reported results.
