On the Bose – Einstein condensate partition function for an ideal gas

Recursive approaches determining the canonical ideal Bose gas partition function are reviewed that enable the Bose – Einstein condensate occupation probability to be calculated for a finite number of particles ensemble where, the thermodynamic limit approximation fails. In addition to the earlier known method recursive in the number of particles, an iteration procedure on the number of quantum states is proposed. The efficiency of both methods is demonstrated for an ideal Bose gas in a three-dimensional isotropic harmonic trap.