The study on the propagation of laser beams of new special modes is a fundamental and significant topic in laser application. Based on the Collins integral formula, the propagation of generalized hypergeometric beams in paraxial ABCD optical systems is studied and the analytical expressions for complex amplitude at output plane are obtained. It is shown that the complex amplitude distribution after propagation in paraxial ABCD optical systems is proportional to Kummer function. By expanding Kummer function as a series, the modulus of this complex amplitude at the output plane could be represented as series summation which shows that the transverse structure of generalized hypergeometric beams propagating in paraxial ABCD optical systems will generally change its form. As a special optical system, the near-field and far-field diffraction behavior in free space is discussed. The transverse structures of both the near-field and far-field are a series of concentric rings and the spatial frequency increases with increase of the radial coordinate. The normalization coefficient was derived and also the relation between simplified modes of hyper-geometric mode and Gaussian modes is analyzed.