On tensor product L-functions for quaternion unitary groups and GL(2) over totally real number fields: Mixed weight cases

We prove an algebraicity of special values of tensor product L-functions for quaternion unitary groups and GL(2) in the mixed weight cases over any totally real number field. As a consequence, we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and GSp(4) when these automorphic forms correspond under the conjectural Jacquet-Langlands correspondence and a period relation for endoscopic liftings of GSp(4) in very general situations.