Measurement of $D_s^{+}$ and $D_s^{*+}$ production in $B$ meson decays and from continuum $e^{+}{e}^{-}$ annihilation at $\sqrt{s}=10.6\mathrm{}$ GeV

New measurements of $D_s^{+}$ and $D_s^{*+}$ meson production rates from B decays and from $q\overline{q}$ continuum events near the ${\rm Y}\mathrm{}\left(4S\right)\mathrm{}$ resonance are presented. Using 20.8 ${\mathrm{fb}}^{\mathrm{-}1}$ of data on the ${\rm Y}\mathrm{}\left(4S\right)\mathrm{}$ resonance and 2.6 ${\mathrm{fb}}^{\mathrm{-}1}$ off-resonance, we find the inclusive branching fractions $\mathcal{B}\left(\overrightarrow{B}{D}_s^{+}X\right)=(10.93\mathrm{}\pm 0.19\pm 0.58\pm 2.73)\%$ and $\mathcal{B}\left(\overrightarrow{B}{D}_s^{*+}X\right)=(7.9\mathrm{}\pm 0.8\pm 0.7\pm 2.0)\%,$ where the first error is statistical, the second is systematic, and the third is due to the $D_s^{+}\to \phi {\pi }^{+}$ branching fraction uncertainty. The production cross sections $\sigma {(e}^{+}{e}^{-}\to D_s^{+}X)\times \mathcal{B}{(D}_s^{+}\to \phi {\pi }^{+})=7.55\mathrm{}\pm 0.20\pm 0.34\mathrm{pb}$ and $\sigma {(e}^{+}{e}^{-}\to D_s^{*\pm }X)\times \mathcal{B}{(D}_s^{+}\to \phi {\pi }^{+})=5.8\mathrm{}\pm 0.7\pm 0.5\mathrm{pb}$ are measured at center-of-mass energies about 40 MeV below the ${\rm Y}\mathrm{}\left(4S\right)\mathrm{}$ mass. The branching fractions $\Sigma \mathcal{B}\left(\overrightarrow{B}{D}_s^{\left(*\right)+}{D}^{\left(*\right)}\right)=(5.07\mathrm{}\pm 0.14\pm 0.30\pm 1.27)\%$ and $\Sigma \mathcal{B}\left(\overrightarrow{B}{D}_s^{*+}{D}^{\left(*\right)}\right)=(4.1\mathrm{}\pm 0.2\pm 0.4\pm 1.0)\%$ are determined from the $D_s^{\left(*\right)+}$ momentum spectra. The mass difference ${m(D}_s^{+}\left)-{m(D}^{+}\right)=98.4\mathrm{}\pm 0.1\pm 0.3\mathrm{MeV}{/c}^2$ is also measured.