  We study the expansion method for the gluon distribution function at low x
values and calculate the charm structure functions in the LO and NLO analysis.
Our results provide a compact formula for the ratio $R^{c}
=F^{c}_{L}/F^{c}_{2}$, which is approximately independent of x and the details
of the parton distribution function at low x values. This ratio could be a good
probe of the charm structure function $F^{c}_{2}$ in the proton deduced from
the reduced charm cross sections at DESY HERA. These results show that the
charm structure functions obtained are in agreement with HERA experimental data
and other theoretical models.
