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 ∆ between the ground state energy ε0 and the first excited state, i.e., ε = ε0 for k = 0 and ε = ε0 + ∆ + csk s , for k > 0, wherehk is the particle momentum and cs a constant with dimensions of energy multiplied by a length to the power s > 0. Explicit formula with arbitrary d/s and ∆ 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 ∆ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at Tc. When ∆ > 0, for any d > 0 exists a Bose-Einstein critical temperature Tc = 0 where 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 functions of d/s. At sufficiently high temperatures ∆-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For ∆ = 0 we recover previous reported results.PACS numbers:

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