  We implement a highly efficient strong-coupling expansion for the Green's
function of the Hubbard model. In the limit of extreme correlations, where the
onsite interaction is infinite, the evaluation of diagrams simplifies
dramatically enabling us to carry out the expansion to the eighth order in
powers of the hopping amplitude. We compute the finite-temperature Green's
function analytically in the momentum and Matsubara frequency space as a
function of the electron density. Employing Pade approximations, we study the
equation of state, Kelvin thermopower, momentum distribution function,
quasiparticle fraction, and quasiparticle lifetime of the system at
temperatures lower than, or of the order of, the hopping amplitude. We also
discuss several different approaches for obtaining the spectral functions
through analytic continuation of the imaginary frequency Green's function, and
show results for the system near half filling. We benchmark our results for the
equation of state against those obtained from a numerical linked-cluster
expansion carried out to the eleventh order.
