  In the paper, we present a calculation of the decay widths for the Higgs
boson decays to the $B_c$, $B_c^*$, $B_c(2^1S_0)$ and $B_c^*(2^3S_1)$ mesons
using the fragmentation-function approach. In the calculation, the
fragmentation functions up to order $\alpha_s^3$ based on the nonrelativistic
QCD factorization theory are used, and the decay widths for $H\to Q+X$ and $H
\to g+X$ at the partonic level are calculated up to order $\alpha_s$. The large
logarithms of $m_H^2/m_{Bc}^2$ are resummed up to next-to-leading logarithmic
accuracy by solving the evolution equations for the running quark masses and
the fragmentation functions. Compared to the leading-order decay widths based
on the nonrelativistic QCD approach, the decay widths based on the
fragmentation-function approach that include the higher-order QCD corrections
are reduced significantly. Our numerical results show that there are about
$1.2\times 10^5$ $B_c$ events via the Higgs decays to be produced at the HL-LHC
with $3ab^{-1}$, and about $1.6\times 10^6$ $B_c$ events via the Higgs decays
to be produced at the HE-LHC with $15ab^{-1}$.
