On the Diophantine equations of the form $$\lambda _1U_{n_1} + \lambda _2U_{n_2} +\cdots + \lambda _kU_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}$$

In this paper, we consider the Diophantine equation $$\lambda _1U_{n_1}+\cdots +\lambda _kU_{n_k}=wp_1^{z_1} \ldots p_s^{z_s},$$ λ 1 U n 1 + ⋯ + λ k U n k = w p 1 z 1 … p s z s , where $$\{U_n\}_{n\ge 0}$$ { U n } n ≥ 0 is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; w is a fixed non-zero integer; $$p_1,\dots ,p_s$$ p 1 , ⋯ , p s are fixed, distinct prime numbers; $$\lambda _1,\dots ,\lambda _k$$ λ 1 , ⋯ , λ k are strictly positive integers; and $$n_1,\dots ,n_k,z_1,\dots ,z_s$$ n 1 , ⋯ , n k , z 1 , ⋯ , z s are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $$(n_1,\dots ,n_k,z_1,\dots ,z_s)$$ ( n 1 , ⋯ , n k , z 1 , ⋯ , z s ) . In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (Monatshefte Math 185(1):103–131, 2018), Mazumdar and Rout (Monatshefte Math 189(4):695–714, 2019), Meher and Rout (Lith Math J 57(4):506–520, 2017), and Ziegler (Acta Arith 190:139–169, 2019).