  Recent breakthroughs in numerical relativity enable one to examine the
validity of the post-Newtonian expansion in the late stages of inspiral. For
the comparison between post-Newtonian (PN) expansion and numerical simulations,
the waveforms in terms of the spin-weighted spherical harmonics are more useful
than the plus and cross polarizations, which are used for data analysis of
gravitational waves. Factorized resummed waveforms achieve better agreement
with numerical results than the conventional Taylor expanded post-Newtonian
waveforms. In this paper, we revisit the post-Newtonian expansion of
gravitational waves for a test-particle of mass $\m$ in circular orbit of
radius $r_0$ around a Schwarzschild black hole of mass $M$ and derive the
spherical harmonic components associated with the gravitational wave
polarizations up to order $v^{11}$ beyond Newtonian. Using the more accurate
$h_{\ell m}$'s computed in this work, we provide the more complete set of
associated $\rho_{\ell m}$'s and $\delta_{\ell m}$'s that form important bricks
in the factorized resummation of waveforms with potential applications for the
construction of further improved waveforms for prototypical compact binary
sources in the future. We also provide ready-to-use expressions of the 5.5PN
gravitational waves polarizations $h_+$ and $h_\times$ in the test-particle
limit for gravitational wave data analysis applications. Additionally, we
provide closed analytical expressions for 2.5PN $h_{\ell m}$, 2PN $\rho_{\ell
m}$ and 3PN $\delta_{\ell m}$, for general multipolar orders $\ell$ and $m$ in
the test-particle limit. Finally, we also examine the implications of the
present analysis for compact binary sources in Laser Interferometer Space
Antenna.
