High-redshift cosmography: Application and comparison with different
  methods
  Cosmography is used in cosmological data processing in order to constrain the
kinematics of the universe in a model-independent way. In this paper, we first
investigate the effect of the ultraviolet (UV) and X-ray relation of a quasar
on cosmological constraints. By fitting the quasar relation and cosmographic
parameters simultaneously, we find that the 4$\sigma$ deviation from the
cosmological constant cold dark matter ($\Lambda$CDM) model disappears. Next,
utilizing the Pantheon sample and 31 long gamma-ray bursts (LGRBs), we make a
comparison among the different cosmographic expansions ($z$-redshift,
$y$-redshift, $E(y)$, $\log(1+z)$, $\log(1+z)+k_{ij}$, and Pad$\rm \acute{e}$
approximations) with the third-order and fourth-order expansions. The expansion
order can significantly affect the results, especially for the $y$-redshift
method. Through analysis from the same sample, the lower-order expansion is
preferable, except the $y$-redshift and $E(y)$ methods. For the $y$-redshift
and $E(y)$ methods, despite adopting the same parameterization of $y=z/(1+z)$,
the performance of the latter is better than that of the former. Logarithmic
polynomials, $\log(1+z)$ and $\log(1+z) + k_{ij}$, perform significantly better
than $z$-redshift, $y$-redshift, and $E(y)$ methods, but worse than Pad$\rm
\acute{e}$ approximations. Finally, we comprehensively analyze the results
obtained from different samples. We find that the Pad$\rm \acute{e}_{(2,1)}$
method is suitable for both low and high redshift cases. The Pad$\rm
\acute{e}_{(2,2)}$ method performs well in a high-redshift situation. For the
$y$-redshift and $E(y)$ methods, the only constraint on the first two
parameters ($q_{0}$ and $j_{0}$) is reliable.
