QCD NLO fragmentation functions for c or b¯ quark to Bc or Bc* meson and their application

The fragmentation functions for a $c$ or $\overline{b}$ quark to a ${B}_{c}$ or ${B}_{c}^{*}$ meson are derived up to QCD next-to-leading order. They are further computed numerically and presented precisely in figures. In order to reach a higher accuracy, we also try to properly use them to estimate ${B}_{c}$ and ${B}_{c}^{*}$ production at a $Z$ factory (an ${e}^{+}{e}^{\ensuremath{-}}$ collider running at the energy of the $Z$-boson pole).