General relativistic quasiequilibrium states of black hole-neutron starbinaries are computed in the moving-puncture framework. We propose threeconditions for determining the quasiequilibrium states and compare thenumerical results with those obtained in the excision framework. We find thatthe results obtained in the moving-puncture framework agree with those in theexcision framework and with those in the third post-Newtonian approximation forthe cases that (i) the mass ratio of the binary is close to unity irrespectiveof the orbital separation, and (ii) the orbital separation is large enough($m_0\Omega \alt 0.02$ where $m_0$ and $\Omega$ are the total mass and theorbital angular velocity, respectively) irrespective of the mass ratio. For$m_0 \Omega \agt 0.03$, both of the results in the moving-puncture and excisionframeworks deviate, more or less, from those in the third post-Newtonianapproximation. Thus the numerical results do not provide a quasicircular state,rather they seem to have a nonnegligible eccentricity of order 0.01--0.1. Weshow by numerical simulation that a method in the moving-puncture framework canprovide approximately quasicircular states in which the eccentricity is by afactor of $\sim 2$ smaller than those in quasiequilibrium given by otherapproaches.
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