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We continue investigation of the universal weight function for the quantumaffine algebra $U_q(\hat{\mathfrak{gl}}_N)$ started in arXiv:math/0610517 andarXiv:0711.2819. We obtain two recurrence relations for the universal weightfunction applying the method of projections developed in arXiv:math/0610398. Onthe level of the evaluation representation of $U_q(\hat{\mathfrak{gl}}_N)$ wereproduce both recurrence relations for the off-shell Bethe vectors calculatedin arXiv:math/0702277 using combinatorial methods.

On the universal weight function for the quantum affine algebra U_q(\hat{\mathfrak{gl}}_N)