Effective results for linear Equations in Members of two Recurrence Sequences

TL;DR

This paper proves that certain linear equations involving two recurrence sequences with dominant roots have only finitely many solutions, and these solutions can be effectively computed, under mild conditions.

Contribution

It establishes finiteness and effective computability of solutions for linear equations in two recurrence sequences with dominant roots, extending previous results.

Findings

Finitely many solutions exist under specified conditions.

Solutions can be explicitly computed.

Applicable to sequences with dominant roots.

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.