On Homogeneous Combinations of Linear Recurrence Sequences

Marie Hubálovská, Štěpán Hubálovský and Eva Trojovská
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Marie Hubálovská: Department of Technical Education, Faculty of Education, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Štěpán Hubálovský: Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Eva Trojovská: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2020, vol. 8, issue 12, 1-7

Abstract: Let ( F n ) n ≥ 0 be the Fibonacci sequence given by F n + 2 = F n + 1 + F n , for n ≥ 0 , where F 0 = 0 and F 1 = 1 . There are several interesting identities involving this sequence such as F n 2 + F n + 1 2 = F 2 n + 1 , for all n ≥ 0 . In 2012, Chaves, Marques and Togbé proved that if ( G m ) m is a linear recurrence sequence (under weak assumptions) and G n + 1 s + ? + G n + ? s ∈ ( G m ) m , for infinitely many positive integers n , then s is bounded by an effectively computable constant depending only on ? and the parameters of ( G m ) m . In this paper, we shall prove that if P ( x 1 , … , x ? ) is an integer homogeneous s -degree polynomial (under weak hypotheses) and if P ( G n + 1 , … , G n + ? ) ∈ ( G m ) m for infinitely many positive integers n , then s is bounded by an effectively computable constant depending only on ? , the parameters of ( G m ) m and the coefficients of P .

Keywords: homogeneous polynomial; linear forms in logarithms; linear recurrence sequence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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