  We compute the short distance coefficients for the NRQCD factorization
formulas of the meson-to-vacuum matrix elements for the axial vector and vector
parts of the charged weak current in the $S$-wave spin-singlet and -triplet
bar{b}c mesons, respectively. The computation is carried out to order alpha_s
including relativistic corrections of all orders in vec{q}^{2n}, where vec{q}
is the relative momentum of the bar{b} and c in the meson rest frame. The
relativistic corrections at order alpha_s are new. The results reveal that the
relativistic corrections to the leptonic decay rate of the B_c meson at order
alpha_s or less converge rapidly, which shows a strong contrast to the
uncomfortably large corrections of order alpha_s^2|vec{q}|^0. The short
distance coefficients listed in this paper can be employed to compute the
resummation of relativistic corrections to the phenomenological measurables
that involve B_c and B_c^* production and decay.
