A generalized super AKNS hierarchy associated with Lie superalgebra sl(2|1) and its super bi-Hamiltonian structure

•Starting from a matrix spectral problem associated with Lie superalgebra sl(2|1), a new generalized super AKNS hierarchy is introduced.•We find that there are four Fermi variables in this new generalized super AKNS hierarchy, which is different from the former researches.•With the help of the super-trace identity, the obtained generalized super AKNS hierarchy can be written as the super bi-Hamiltonian structure.