Power spectra and spectral indices of $k$-inflation: high-order
  corrections
  $k$-inflation represents the most general single-field inflation, in which
the perturbations usually obey an equation of motion with a time-dependent
sound speed. In this paper, we study the observational predictions of the
$k$-inflation by using the high-order uniform asymptotic approximation method.
We calculate explicitly the slow-roll expressions of the power spectra,
spectral indices, and running of the spectral indices for both the scalar and
tensor perturbations. These expressions are all written in terms of the Hubble
and sound speed flow parameters. It is shown that the previous results obtained
by using the first-order uniform asymptotic approximation have been
significantly improved by the high-order corrections of the uniform asymptotic
approximations. Furthermore, we also check our results by comparing them with
the ones obtained by other approximation methods, including the Green's
function method, WKB approximation, and improved WKB approximation, and find
the relative errors.
