Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression

Kunle Adegoke, Robert Frontczak and Taras Goy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 54—63
DOI: 10.7546/nntdm.2021.27.2.54-63
Download full paper: PDF, 204 Kb

Details

Authors and affiliations

Kunle Adegoke
Department of Physics and Engineering Physics, Obafemi Awolowo University
220005 Ile-Ife, Nigeria

Robert Frontczak
Landesbank Baden-Württemberg
70173 Stuttgart, Germany

Taras Goy
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
76018 Ivano-Frankivsk, Ukraine

Abstract

We evaluate the partial sum of the products of the terms of any two Horadam sequences with subscripts in arithmetic progression. Illustrative examples are drawn from six well-known Horadam sequences.

Keywords

  • k-generalized Lucas numbers
  • Linear form in logarithms
  • Reduction method

2020 Mathematics Subject Classification

  • 11B39
  • 11J86

References

  1. Adegoke, K. (2018). Partial sums and generating functions for power of second order sequences with indices in arithmetic progression. Preprint. Available online at: https://arxiv.org/abs/1904.09916.
  2. Belbachir, H., & Bencherif, F. (2013). Sums of products of generalized Fibonacci and Lucas numbers. Ars Combinatoria, 110, 33–43.
  3. Čerin, Z. (2015). Formulas for quadratic sums that involve generalized Fibonacci and Lucas numbers. Rad Hrvatske Akademije Znanosti i Umjetnosti, Matematicke Znanosti, 19(523), 1–12.
  4. Čerin, Z. (2009). On sum of products of Horadam numbers. Kyungpook Mathematical Journal, 49, 483–492.
  5. Čerin, Z. (2009) Sum of products of generalized Fibonacci and Lucas numbers.
    Demonstratio Mathematica, 42, 247–258.
  6. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
  7. Khan, M. A., & Kwong, H. (2014). On sums of products of Fibonacci-type recurrences. The Fibonacci Quarterly, 52(2), 20–26.
  8. Kiliç, E., Ömür, N., & Ulutas, Y. T. (2011). Some finite sums involving generalized Fibonacci and Lucas numbers. Discrete Dynamics in Nature and Society, 2011(1), Article ID 284261.
  9. Kiliç, E., & Prodinger, H. (2015). Sums of products of generalized Fibonacci and Lucas numbers. Acta Mathematica Hungarica, 145(1), 17–25.
  10. Kiliç, E., & Stănică, P. (2013). General approach in computing sums of products of binary sequences. Hacettepe Journal of Mathematics and Statistics, 42(1), 1–7.
  11. Koshy, T. (2017). Fibonacci and Lucas Numbers with Applications, Wiley, New York.
  12. Koshy, T. (2014). Pell and Pell–Lucas Numbers with Applications, Springer, New York.
  13. Larcombe, P. J. (2017). Horadam sequences: A survey update and extension. Bulletin of the Institute of Combinatorics and its Applications, 80, 99–118.
  14. Ribenboim, P. (2000). My Numbers, My Friends, Springer, New York.
  15. Sloane, N. J. A. (ed.). The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org.
  16. Vajda, S. (2008). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications, Dover, New York.

Related papers

Cite this paper

Adegoke, K., Frontczak, R., & Goy, T. (2021). Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression. Notes on Number Theory and Discrete Mathematics, 27(2), 54-63, doi: 10.7546/nntdm.2021.27.2.54-63.

Comments are closed.