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Polynomials whose coefficients are generalized Leonardo numbers

  • Mark Shattuck
Published/Copyright: May 24, 2024
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 74 Issue 2

Abstract

Let an be defined recursively by an = an–1 + an–2 + pn + q if n ≥ 2, with a0 = a1 = 1, where p and q are constants. The p = 0 case of an corresponds to what have been termed the generalized Leonardo numbers, the q = 1 case of which are the classical Leonardo numbers. Let Ln(x)=∑i=0naixn−i, which we refer to as a generalized Leonardo coefficient polynomial. Here, it is shown, under some mild assumptions on p and q, that Ln(x) has exactly one real zero if n is odd and no real zeros if n is even. This yields a previous result for the Fibonacci coefficient polynomials when p = q = 0 and a new analogous result for the Leonardo coefficient polynomials when p = 0, q = 1. Further, when p = 0 and –1 < q ≤ 1 + ρ, we show that the sequence consisting of the real zeros of the Ln(x) for n odd is strictly decreasing with limit –ρ, where ρ=1+52. Finally, under the same assumptions on p and q, we are able to show the convergence in modulus of all the zeros of Ln(x).

MSC 2010: Primary 11C08; 13B25
Keywords: Leonardo numbers; zeros of polynomials; linear recurrences; Rouché’s Theorem

  1. Communicated by István Gaál

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Received: 2023-05-18
Accepted: 2023-08-10
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Right algebras in Sup and the topological representation of semi-unital and semi-integral quantales, revisited
  2. Topological representation of some lattices
  3. Exact-m-majority terms
  4. Polynomials whose coefficients are generalized Leonardo numbers
  5. A study on error bounds for Newton-type inequalities in conformable fractional integrals
  6. Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
  7. Close-to-convex functions associated with a rational function
  8. Complete monotonicity for a ratio of finitely many gamma functions
  9. Class of bounds of the generalized Volterra functions
  10. Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
  11. Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line
  12. Solving Fredholm integro-differential equations involving integral condition: A new numerical method
  13. Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation
  14. Weighted 1MP and MP1 inverses for operators
  15. Non-commutative effect algebras, L-algebras, and local duality
  16. Operator Bohr-type inequalities
  17. Some results for weighted Bergman space operators via Berezin symbols
  18. Lower separation axioms in bitopogenous spaces
  19. Fisher information in order statistics and their concomitants for Cambanis bivariate distribution
  20. Irreducibility of strong size levels
https://doi.org/10.1515/ms-2024-0023
Keywords for this article
Leonardo numbers; zeros of polynomials; linear recurrences; Rouché’s Theorem

Articles in the same Issue

  1. Right algebras in Sup and the topological representation of semi-unital and semi-integral quantales, revisited
  2. Topological representation of some lattices
  3. Exact-m-majority terms
  4. Polynomials whose coefficients are generalized Leonardo numbers
  5. A study on error bounds for Newton-type inequalities in conformable fractional integrals
  6. Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection
  7. Close-to-convex functions associated with a rational function
  8. Complete monotonicity for a ratio of finitely many gamma functions
  9. Class of bounds of the generalized Volterra functions
  10. Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
  11. Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line
  12. Solving Fredholm integro-differential equations involving integral condition: A new numerical method
  13. Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation
  14. Weighted 1MP and MP1 inverses for operators
  15. Non-commutative effect algebras, L-algebras, and local duality
  16. Operator Bohr-type inequalities
  17. Some results for weighted Bergman space operators via Berezin symbols
  18. Lower separation axioms in bitopogenous spaces
  19. Fisher information in order statistics and their concomitants for Cambanis bivariate distribution
  20. Irreducibility of strong size levels
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