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 $$a_1U_{n_1}+\cdots +a_k U_{n_k}=b_1V_{m_1}+\cdots + b_\ell V_{m_\ell}$$ 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.