Effective results for linear Equations in Members of two Recurrence Sequences

Abstract

Let $(U_n{)}_{n=0}^{\infty }$ and $(V_m{)}_{m=0}^{\infty }$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots ,a_k)\in {\mathbb{Z}}^k$ and fixed $\ell$-tuple $(b_1,\dots ,b_{\ell })\in {\mathbb{Z}}^{\ell }$ we consider the linear equation $$a_1{U}_{n_1}+\cdots +a_k{U}_{n_k}=b_1{V}_{m_1}+\cdots +b_{\ell }{V}_{m_{\ell }}$$ in the unknown non-negative integers $n_1,\dots ,n_k$ and $m_1,\dots ,m_{\ell }$. Under the assumption that the linear recurrences $(U_n{)}_{n=0}^{\infty }$ and $(V_m{)}_{m=0}^{\infty }$ have dominant roots and under the assumption of further mild restrictions we show that this equation has only finitely many solutions which can be found effectively.