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We consider new modes of two-neutrino and neutrinoless double-β decays in which one β electron goes over to a continuous spectrum and the other occupies a vacant bound level of the daughter ion. We calculate the corresponding phase-space factors of the final states, estimate the partial decay rates, and derive the one- and two-electron energy spectra using relativistic many-electron wave functions of atoms provided by the multiconfiguration Dirac-Hartree-Fock package GRASP2K. While thearXiv:1805.07815v1 fatcat:xxbaqztrbfghnn6d56mv5oxbui