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On Homogeneous Combinations of Linear Recurrence Sequences

Author

Listed:
  • 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)

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 + ℓ s ∈ ( G m ) m , for infinitely many positive integers n , then s is bounded by an effectively computable constant depending only on ł and the parameters of ( G m ) m . In this paper, we shall prove that if P ( x 1 , … , x ℓ ) is an integer homogeneous s -degree polynomial (under weak hypotheses) and if P ( G n + 1 , … , G n + ℓ ) ∈ ( G m ) m for infinitely many positive integers n , then s is bounded by an effectively computable constant depending only on ℓ , the parameters of ( G m ) m and the coefficients of P .

Suggested Citation

  • Marie Hubálovská & Štěpán Hubálovský & Eva Trojovská, 2020. "On Homogeneous Combinations of Linear Recurrence Sequences," Mathematics, MDPI, vol. 8(12), pages 1-7, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2152-:d:455477
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    References listed on IDEAS

    as
    1. Pavel Trojovský, 2019. "On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes," Mathematics, MDPI, vol. 7(8), pages 1-10, August.
    2. Pavel Trojovský, 2020. "Fibonacci Numbers with a Prescribed Block of Digits," Mathematics, MDPI, vol. 8(4), pages 1-7, April.
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