Theoretically we study the doubly resonant IR-UV hyper-Raman scattering where the IR light is resonant to the vibrational transition and the UV/visible light is resonant to the electronic transition between the ground and excited states. Based on the Taylor expansion of the electric transition dipole moments with respect to the normal coordinates, we have derived the expressions for the hyper-Raman A, B, and C terms. Using quantum chemistry calculations, we have estimated the magnitudes for all the three terms. Due to double resonance, contributions from all the three terms should be detectable in experiments.