Jacobi quasi-Nijenhuis Algebroids

In this paper, for a Jacobi algebroid $A$, by introducing the notion of Jacobi quasi-Nijenhuis algebroids, which is a generalization of Poisson quasi-Nijenhuis manifolds introduced by Sti'{e}non and Xu, we study generalized complex structures on the Courant-Jacobi algebroid $A\oplus A^{\ast }$, which unifies generalized complex (contact) structures on an even(odd)-dimensional manifold.