We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians Y (gl(m|n) in terms of the current generators of the Yangian double DY (gl(m|n)). We use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double DY (gl(m|n)) we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra U q ( gl m ). We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients.In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by gl(m|n) superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.