On Homogeneous Combinations of Linear Recurrence Sequences

Autor: Marie Hubálovská, Štěpán Hubálovský, Eva Trojovská
Jazyk: angličtina
Rok vydání: 2020
Předmět:
Zdroj: Mathematics, Vol 8, Iss 12, p 2152 (2020)
Druh dokumentu: article
ISSN: 2227-7390
DOI: 10.3390/math8122152
Popis: Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2=Fn+1+Fn, for n≥0, where F0=0 and F1=1. There are several interesting identities involving this sequence such as Fn2+Fn+12=F2n+1, for all n≥0. In 2012, Chaves, Marques and Togbé proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and Gn+1s+⋯+Gn+ℓs∈(Gm)m, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of (Gm)m. In this paper, we shall prove that if P(x1,…,xℓ) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(Gn+1,…,Gn+ℓ)∈(Gm)m for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of (Gm)m and the coefficients of P.
Databáze: Directory of Open Access Journals
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