  We theoretically propose and investigate a feasible experimental scheme for
the realization of the dynamical Casimir effect (DCE) in a hybrid
optomechanical cavity with a moving end mirror containing an interacting
cigar-shaped Bose-Einstein condensate (BEC). We show that in the red-detuned
regime of cavity optomechanics together with the weak optomechanical coupling
limit by \textit{coherent} modulation of the \textit{s}-wave scattering
frequency of the BEC and the mechanical spring coefficient of the mechanical
oscillator (MO), the mechanical and atomic quantum vacuum fluctuations are
parametrically amplified, which consequently lead to the generation of the
mechanical/Bogoliubov-type Casimir phonons. Interestingly, in the coherent
regime corresponding to the case of largely different optomechanical coupling
strengths of the cavity field to the BEC and the MO, or equivalently largely
different cooperativities, one can generate a large number of Casimir photons
due to the amplification of the intracavity vacuum fluctuations
\textit{induced} by the time modulations of the BEC and the MO. The number of
generated Casimir particles are externally controllable by the cooperativities,
and the modulation amplitudes of the atomic collisions rate and the mechanical
spring coefficient.
