We perform in this work, through the techniques of transfer matrix, an analyticalstudy of the photonic band structure and the density of states of a one dimensionalphotonic crystal consisting of a super-lattice composed by two alternating layers of air andGaAs of thicknesses a and the b, respectively. We find photonic band gaps, as expected,and also null gap points. We study the reflectivity and the transmissivity of theelectromagnetic field through the structure. We further analyze the symmetry properties ofthe solutions for the electromagnetic field and find that, in general, these solutions have awell defined parity except at the null gap points where there are degeneracies. At thesepoints we show that the phase difference between the various waves, reflected at allinterfaces, interfere destructively so that, in contrast with the Bragg mirrors, the incidentwave is efficiently transmitted through the structure.