2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142480We continue investigation of the universal weight function for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ started in arXiv:math/0610517 and arXiv:0711.2819. We obtain two recurrence relations for the universal weight function applying the method of projections developed in arXiv:math/0610398. On the level of the evaluation representation of $U_q(\hat{\mathfrak{gl}}_N)$ we reproduce both recurrence relations for the off-shell Bethe vectors calculated in arXiv:math/0702277 using combinatorial methods.35 pages, typos correctedQuantum AlgebraHigh Energy Physics - TheoryMathematical PhysicsExactly Solvable and Integrable Systems17B37; 16W30; 81R50; 82B23text