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Generalized Bose-Einstein (BE) and Fermi-Dirac (FD) distributions innonextensive quantum statistics have been discussed by the maximum-entropymethod (MEM) with the optimum Lagrange multiplier based on the exact integralrepresentation [Rajagopal, Mendes, and Lenzi, Phys. Rev. Lett. {\bf 80}, 3907(1998)]. It has been shown that the $(q-1)$ expansion in the exact approachagrees with the result obtained by the asymptotic approach valid for $O(q-1)$.Model calculations have been made with a uniform density of states forelectrons and with the Debye model for phonons. Based on the result of theexact approach, we have proposed the {\it interpolation approximation} to thegeneralized distributions, which yields results in agreement with the exactapproach within $O(q-1)$ and in high- and low-temperature limits. By using thefour methods of the exact, interpolation, factorization and superstatisticalapproaches, we have calculated coefficients in the generalized Sommerfeldexpansion, and electronic and phonon specific heats at low temperatures. Acomparison among the four methods has shown that the interpolationapproximation is potentially useful in the nonextensive quantum statistics.Supplementary discussions have been made on the $(q-1)$ expansion of thegeneralized distributions based on the exact approach with the use of theun-normalized MEM, whose results also agree with those of the asymptoticapproach.

Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches