Ultra-massive white dwarfs are relevant for their role as type Ia Supernova progenitors, the occurrence of physical processes in the asymptotic giant-branch phase, the existence of high-field magnetic white dwarfs, and the occurrence of double white dwarf mergers. Some hydrogen-rich ultra-massive white dwarfs are pulsating stars, and as such, they offer the possibility of studying their interiors through asteroseismology. On the other hand, pulsating helium-rich ultra-massive white dwarfs could be even more attractive objects for asteroseismology if they were found, as they should be hotter and less crystallized than pulsating hydrogen-rich white dwarfs, something that would pave the way for probing their deep interiors. We explore the pulsational properties of ultra-massive helium-rich white dwarfs with carbon-oxygen and oxygen-neon cores resulting from single stellar evolution. Our goal is to provide a theoretical basis that could eventually help to discern the core composition of ultra-massive white dwarfs and the scenario of their formation through asteroseismology, anticipating the possible future detection of pulsations in this type of stars. We find that, given that the white dwarf models coming from the three scenarios considered are characterized by distinct core chemical profiles, their pulsation properties are also different, thus leading to distinctive signatures in the period-spacing and mode-trapping properties. Our results indicate that, in case of an eventual detection of pulsating ultra-massive helium-rich white dwarfs, it would be possible to derive valuable information encrypted in the core of these stars in connection with the origin of such exotic objects. The detection of pulsations in these stars has many chances to be achieved soon through observations collected with ongoing space missions.
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