A recent reexamination of BCS theory leads one to devise a
generalized BEC formalism (GBEC) that is essentially a boson-fermion (BF)
gas model containing three new ingredients: i) Cooper pairs (CPs),
in contrast to BCS pairs, as real bosons; ii) BF vertex interactions
(analogous to electron-phonon vertex interactions in metals) and
iii) two-hole CPs (2hCPs) along with two-electron CPs (2eCPs) on an
equal footing. In addition to the usual normal phase at very high
temperatures T, three condensed phases emerge at low T: two pure
BECs of 2eCPs and of 2hCPs, as well as a mixed phase of both kinds of CPs.
BCS theory is precisely recovered with a mixed phase of equal numbers
of both kinds of CPs. Substantially higher Tcs than BCS theory are
predicted and, in contrast to the well-known BCS exponential rise of Tc
from zero, the GBEC scheme exhibits a linear rise as function of doping
as eminently observed in high-Tc superconductors. Ultracold fermionic
atoms are briefly discussed, especially around the so-called unitarity
range where the two-fermion scattering length can diverge.