  We derive gluon fragmentation functions in the Nambu-Jona-Lasinio (NJL) model
by treating a gluon as a pair of color lines formed by fictitious quark and
anti-quark ($q\bar q$). Gluon elementary fragmentation functions are obtained
from the quark and anti-quark elementary fragmentation functions for emitting
specific mesons in the NJL model under the requirement that the $q\bar q$ pair
maintains in the flavor-singlet state after meson emissions. An integral
equation, which iterates the gluon elementary fragmentation functions to all
orders, is then solved to yield the gluon fragmentation functions at a model
scale. It is observed that these solutions are stable with respect to variation
of relevant model parameters, especially after QCD evolution to a higher scale
is implemented. We show that the inclusion of the gluon fragmentation functions
into the theoretical predictions from only the quark fragmentation functions
greatly improves the agreement with the SLD data for the pion and kaon
productions in $e^+e^-$ annihilation. Our proposal provides a plausible
construct for the gluon fragmentation functions, which are supposed to be null
in the NJL model.
