  Inspired by the recent measurements of the $\eta_c$ meson production at LHC,
we investigate the relativistic correction effect for the fragmentation
function of the gluon into $\eta_c$, which constitutes the crucial
nonperturbative elements to understand $\eta_c$ production at high $p_T$.
Employing three distinct methods, we calculate the leading relativistic
correction to the $g\to\eta_c$ fragmentation function in the NRQCD
factorization framework, as well as verify the existing NLO result for the
$c\to \eta_c$ fragmentation function. We also study the evolution behavior of
these fragmentation functions with the aid of DGLAP equation.
