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The fragmentation functions for $c$ or $\bar{b}$ quark to $B_c$ or $B_c^*$meson are derived up to QCD next-to-leading order (NLO). They are furthercomputed numerically and presented in figures precisely. In order to reach ahigher accuracy, we have also tried properly to apply them to estimating the$B_c$ and $B_c^*$ production at a $Z$-factory (an $e^+e^-$ collider running atthe energy of the $Z$-boson pole).

The QCD NLO fragmentation functions for $c$ or $\bar{b}$ quark to $B_c$ or $B_c^*$ meson and their application