S-parts of sums of terms of linear recurrence sequences

Let $$S= \{ p_1, \ldots, p_s\}$$ be a finite, non-empty set of distinct prime numbers and $$(U_{n})_{n \geq 0}$$ linear recurrence sequence integers order at least 2. For any positive integer k, $$w = (w_k, w_1)\in\mathbb{Z}^k, w_1, w_k\neq 0$$ we define $$(U_j^{(k, w)})_{j\geq 1}$$ an increasing composed the form $$|w_kU_{n_k} +\cdots + w_1U_{n_1}|$$ , $$ n_k>\cdots >n_1$$ . Under certain assumptions, prove that for $$\varepsilon >0$$ there exists $$n_{0}$$ such $$[U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},$$ $${\rm for}\, j > n_0$$ where $$[m]_S$$ denotes S-part m. On further assumptions on also compute effective bound $$[U_j^{(k, w)}]_S$$ $$(U_j^{(k,w)})^{1-c}$$ c is constant depending only r, $$a_1$$ ., $$a_r$$ $$U_0$$ $$U_{r-1}$$ $$w_1$$ $$w_k$$ S.