  We revisit the cut-off prescriptions which are needed in order to specify
completely the form of Tsallis' maximum entropy distributions. For values of
the Tsallis entropic parameter $q>1$ we advance an alternative cut-off
prescription and discuss some of its basic mathematical properties. As an
illustration of the new cut-off prescription we consider in some detail the
$q$-generalized quantum distributions which have recently been shown to
reproduce various experimental results related to high $T_c$ superconductors.
