  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 $U_q(\widehat{\mathfrak{gl}}_{n})$. 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.
