  We investigate unpolarized and polarized gluon distributions and their
applications to the Ioffe-time distributions, which are related to lattice QCD
calculations of parton distribution functions. Guided by the counting rules
based on the perturbative QCD at large momentum fraction $x$ and the color
coherence of gluon couplings at small $x$, we parametrize gluon distributions
in the helicity basis. By fitting the unpolarized gluon distribution, the
inferred polarized gluon distribution from our parametrization agrees with the
one from global analysis. A simultaneous fit to both unpolarized and polarized
gluon distributions is also performed to explore the model uncertainty. The
agreement with the global analysis supports the $(1-x)$ power suppression of
the helicity-antialigned distribution relative to the helicity-aligned
distribution. The corresponding Ioffe-time distributions and their asymptotic
expansions are calculated from the gluon distributions. Our results of the
Ioffe-time distributions can provide guidance to the extrapolation of lattice
QCD data to the region lacking precise gluonic matrix elements. Therefore, they
can help regulate the ill-posed inverse problem associated with extracting the
gluon distributions from discrete data from first-principle calculations, which
are available in a limited range of the nucleon momentum and the spatial
separation between the gluonic currents. Given various limitations in obtaining
lattice QCD data at large Ioffe time, phenomenological approaches can provide
important complementary information to extract the gluon distributions in the
entire $x$ region. The possibility of investigating higher-twist effects and
other systematic uncertainties in the contemporary first-principle calculations
of parton distributions from phenomenologically well-determined Ioffe-time
distributions in the large Ioffe-time region is also discussed.
