  We explore the spectral problem of the two photon Rabi model from the point
of view of complex differential equations in the Bargmann representation. The
wave-functions are automatically entire but to ensure finite norm one has to
effectively construct asymptotic expansions. This is achieved by means of the
Mellin transformation and convergent factorial series, which allow direct
computation of the spectral determinant. By further analysing the differential
equation satisfied by the Mellin transform, we obtain a new form of the
spectral conditions -- in terms of holonomy matrices and contour integrals.
