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 \(\lambda _1U_{n_1}+\cdots +\lambda _kU_{n_k}=wp_1^{z_1} \ldots p_s^{z_s},\) ${\lambda }_1{U}_{n_1}+\cdots +{\lambda }_k{U}_{n_k}=w{p}_1^{z_1}\dots p_s^{z_s},$ where \(\{U_n\}_{n\ge 0}\) ${\{U_n\}}_{n\ge 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; w is a fixed non-zero integer; \(p_1,\dots ,p_s\) $p_1,\cdots ,p_s$ are fixed, distinct prime numbers; \(\lambda _1,\dots ,\lambda _k\) ${\lambda }_1,\cdots ,{\lambda }_k$ are strictly positive integers; and \(n_1,\dots ,n_k,z_1,\dots ,z_s\) $n_1,\cdots ,n_k,z_1,\cdots ,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions \((n_1,\dots ,n_k,z_1,\dots ,z_s)\) $(n_1,\cdots ,n_k,z_1,\cdots ,z_s)$ . 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).