Based on the important position and application of abnormal Boson quantum system in the theory of relativistic covariant vector field of zero quality, the specific form of eigenket of the abnormal Boson creation operator can be derived by virtue of the contour integral representation of Dirac-function and its properties as well as feature of the number representation of abnormal Boson system. Furthermore, two sets of new orthogonality and completeness relations, which are irrelevant to normality operator, were obtained. Finally, as an applications, it was used to rewrite the expressions of Hermite polynomial and Laguerre polynomial under meaning of the contour integral, which presumably provide help for some questions of quantum optics about these polynomials.