The thermal behaviors of the order-parameters, susceptibility and specific heat of the mixed spin-1/2 and spin-1 Blume-Capel Ising ferrimagnetic system is studied on the Bethe lattice by using the exact recursion equations. The exact equations for the order parameters and the susceptibility are obtained for the sublattices with spin-1 and spin-1/2. Also, the exact expressions for the free energy and the specific heat are obtained on the Bethe lattice as a whole system. Then the thermal behaviors of the order-parameters, susceptibility, inverse susceptibility and specific heat are studied for the coordination numbers q = 4 and q = 6. The obtained results are illustrated and discussed in detail.