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We obtain recursion formulas for the Bethe vectors of models with periodicboundary conditions solvable by the nested algebraic Bethe ansatz and based onthe quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also presenta sum formula for their scalar products. This formula describes the scalarproduct in terms of a sum over partitions of the Bethe parameters, whosefactors are characterized by two highest coefficients. We provide differentrecursions for these highest coefficients. In addition, we show that when theBethe vectors are on-shell, their norm takes the form of a Gaudin determinant.

Scalar products and norm of Bethe vectors for integrable models based on $U_q(\widehat{\mathfrak{gl}}_{n})$