Diophantine equations related to reciprocals of linear recurrence sequences

H. R. Hashim and Sz. Tengely
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 49-56
DOI: 10.7546/nntdm.2019.25.2.49-56
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Authors and affiliations

H. R. Hashim
Mathematical Institute, University of Debrecen
P. O. Box 12, 4010 Debrecen, Hungary

Sz. Tengely
Mathematical Institute, University of Debrecen
P. O. Box 12, 4010 Debrecen, Hungary

Abstract

In the past years, many researchers have worked on degenerate polynomials and a variety of its extentions and variants can be found in literature. Following up, in this article, we
firstly introduce the partially degenerate Legendre–Genocchi polynomials, and further define a new generalization of degenerate Legendre–Genocchi polynomials. From our generalization, we establish some implicit summation formulae and symmetry identities by the generating function of partially degenerate Legendre–Genocchi polynomials. Eventually, it can be found that some recently demonstrated summations and identities stated in the article, are special cases of our results.

Keywords

  • Recurrence sequences0
  • Diophantine equations.

2010 Mathematics Subject Classification

  • 11D25
  • 11B39

References

  1. Bravo, J. J. & Luca, F. (2016). On the Diophantine equation Fn + Fm = 2a, Quaest. Math., 39 (3), 391–400.
  2. Chim, K. C. & Ziegler, V. (2018). On Diophantine equations involving sums of Fibonacci numbers and powers of 2, Integers 18, Article A99, 30.
  3. Hashim, H. R. & Tengely, Sz. (2018). Representations of reciprocals of Lucas sequences. Miskolc Mathematical Notes, 19 (2), 865–872.
  4. Köhler, G. (1985). Generating functions of Fibonacci-like sequences and decimal expansions of some fractions. Fibonacci Quart., 23 (1), 29–35.
  5. Luca, F. (2000). Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math., 57 (2), 243–254.
  6. Runge, C. (1887). Uber ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math., 100, 425–435.
  7. Stancliff, F. (1953). A curious property of aii. Scripta Math., 19, 126.
  8. Stein, W. A. et al. (2019). Sage Mathematics Software (Version 8.6). The Sage Development Team. Available online: http://www.sagemath.org.
  9. Tengely, Sz. (2015). On the Lucas sequence equation 1/Un = ∑k=1 Uk-1 / xk. Period. Math. Hung., 71 (2), 236–242.
  10. de Weger, B. M. M. (1995). A curious property of the eleventh Fibonacci number. Rocky Mountain J. Math., 25 (3), 977–994.

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Cite this paper

APA

Hashim, H. R. & Tengely, Sz. (2019). Diophantine equations related to reciprocals of linear recurrence sequences. Notes on Number Theory and Discrete Mathematics, 25(2), 49-56, doi: 10.7546/nntdm.2019.25.2.49-56.

Chicago

Hashim, H. R. and Sz. Tengely “Diophantine equations related to reciprocals of linear recurrence sequences.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 49-56, doi: 10.7546/nntdm.2019.25.2.49-56.

MLA

Hashim, H. R. and Sz. Tengely. “Diophantine equations related to reciprocals of linear recurrence sequences.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 49-56. Print, doi: 10.7546/nntdm.2019.25.2.49-56.

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