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Year 2025, , 65 - 71, 26.06.2025

Yasemin Alp

https://doi.org/10.36753/mathenot.1607457

Abstract

References

  • [1] Koshy, T.: Fibonacci and Lucas Numbers with Applications. John Wiley & Sons, 2001.
  • [2] Vajda, S.: Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Halsted Press, 1989.
  • [3] Catalini, M.: Generalized bivariate Fibonacci polynomials. arxiv:0211366. (2002).
  • [4] Tuglu, N., Kocer, E. G., Stakhov, A.: Bivariate Fibonacci like p-polynomials. Applied Mathematics and Computation. 217(24), 10239-10246 (2011).
  • [5] Kocer, E.G., Tuglu, N., Stakhov, A.: On the m-extension of the Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals. 40(4) 1890–1906 (2009).
  • [6] Gould, H.W.: Series transformations for finding recurrences for sequences. The Fibonacci Quarterly. 28(2), 166-71 (1990).
  • [7] Haukkanen, P.: Formal power series for binomial sums of sequences of numbers. The Fibonacci Quarterly. 31(1), 28–31 (1993).
  • [8] Spivey, M. Z., Steil, L. L.: The k-binomial transforms and the Hankel transform. Journal of Integer Sequences. 9(1), 19 (2006).
  • [9] Chen, K.W.: Identities from the binomial transform. Journal of Number Theory. 124(1), 142-150 (2007).
  • [10] Prodinger, H.: Some information about the binomial transform. The Fibonacci Quarterly. 32(5), 412–415 (1994).
  • [11] Falcón, S.: Binomial transform of the generalized k-Fibonacci numbers. Communications in Mathematics and Applications, 10(3), 643–651 (2019).
  • [12] Falcon, S., Plaza, A.: Binomial transforms of the k-Fibonacci sequence. International Journal of Nonlinear Sciences and Numerical Simulation. 10(11-12), 1527-1538 (2009).
  • [13] Kwon, Y.: Binomial transforms of the modified k-Fibonacci-like sequence, International Journal of Mathematics and Computer Science. 14(1), 47–59 (2019).
  • [14] Yilmaz, N.: Binomial transforms of the balancing and Lucas-balancing polynomials. Contributions to Discrete Mathematics. 15(3), 133-144 (2020).
  • [15] Bhadouria, P., Jhala, D., Singh, B.: Binomial transforms of the k-Lucas sequences and its properties. Journal of Mathematics and Computer Science. 8, 81–92 (2014).
  • [16] Yilmaz, N., Aktaş, I.: Special transforms of the generalized bivariate Fibonacci and Lucas polynomials. Hacettepe Journal of Mathematics and Statistics. 52(3), 640-651 (2023).
  • [17] Yilmaz, N., Taskara, N.: Binomial transforms of the Padovan and Perrin matrix sequences. Abstract and Applied Analysis. 2013(1), (2013).

The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials

Year 2025, , 65 - 71, 26.06.2025

Yasemin Alp

https://doi.org/10.36753/mathenot.1607457

Abstract

The generalized binomial transforms of the bivariate Fibonacci $p-$polynomials and Lucas $p-$polynomials are introduced in this study. Furthermore, the generating functions of these polynomials are provided. Moreover, some relations are found for them. All results obtained are reduced to the $k-$binomial, falling binomial, rising binomial, and binomial transforms of the Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Fibonacci, and Lucas numbers.

Keywords

Binomial transform Bivariate Fibonacci $p-$polynomials Bivariate Lucas $p-$polynomials Generating function

References

  • [1] Koshy, T.: Fibonacci and Lucas Numbers with Applications. John Wiley & Sons, 2001.
  • [2] Vajda, S.: Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Halsted Press, 1989.
  • [3] Catalini, M.: Generalized bivariate Fibonacci polynomials. arxiv:0211366. (2002).
  • [4] Tuglu, N., Kocer, E. G., Stakhov, A.: Bivariate Fibonacci like p-polynomials. Applied Mathematics and Computation. 217(24), 10239-10246 (2011).
  • [5] Kocer, E.G., Tuglu, N., Stakhov, A.: On the m-extension of the Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals. 40(4) 1890–1906 (2009).
  • [6] Gould, H.W.: Series transformations for finding recurrences for sequences. The Fibonacci Quarterly. 28(2), 166-71 (1990).
  • [7] Haukkanen, P.: Formal power series for binomial sums of sequences of numbers. The Fibonacci Quarterly. 31(1), 28–31 (1993).
  • [8] Spivey, M. Z., Steil, L. L.: The k-binomial transforms and the Hankel transform. Journal of Integer Sequences. 9(1), 19 (2006).
  • [9] Chen, K.W.: Identities from the binomial transform. Journal of Number Theory. 124(1), 142-150 (2007).
  • [10] Prodinger, H.: Some information about the binomial transform. The Fibonacci Quarterly. 32(5), 412–415 (1994).
  • [11] Falcón, S.: Binomial transform of the generalized k-Fibonacci numbers. Communications in Mathematics and Applications, 10(3), 643–651 (2019).
  • [12] Falcon, S., Plaza, A.: Binomial transforms of the k-Fibonacci sequence. International Journal of Nonlinear Sciences and Numerical Simulation. 10(11-12), 1527-1538 (2009).
  • [13] Kwon, Y.: Binomial transforms of the modified k-Fibonacci-like sequence, International Journal of Mathematics and Computer Science. 14(1), 47–59 (2019).
  • [14] Yilmaz, N.: Binomial transforms of the balancing and Lucas-balancing polynomials. Contributions to Discrete Mathematics. 15(3), 133-144 (2020).
  • [15] Bhadouria, P., Jhala, D., Singh, B.: Binomial transforms of the k-Lucas sequences and its properties. Journal of Mathematics and Computer Science. 8, 81–92 (2014).
  • [16] Yilmaz, N., Aktaş, I.: Special transforms of the generalized bivariate Fibonacci and Lucas polynomials. Hacettepe Journal of Mathematics and Statistics. 52(3), 640-651 (2023).
  • [17] Yilmaz, N., Taskara, N.: Binomial transforms of the Padovan and Perrin matrix sequences. Abstract and Applied Analysis. 2013(1), (2013).
There are 17 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Yasemin Alp 0000-0002-4146-7374

Early Pub Date May 19, 2025
Publication Date June 26, 2025
Submission Date December 25, 2024
Acceptance Date February 12, 2025
Published in Issue Year 2025

Cite

APA Alp, Y. (2025). The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Mathematical Sciences and Applications E-Notes, 13(2), 65-71. https://doi.org/10.36753/mathenot.1607457
AMA Alp Y. The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Math. Sci. Appl. E-Notes. June 2025;13(2):65-71. doi:10.36753/mathenot.1607457
Chicago Alp, Yasemin. “The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials”. Mathematical Sciences and Applications E-Notes 13, no. 2 (June 2025): 65-71. https://doi.org/10.36753/mathenot.1607457.
EndNote Alp Y (June 1, 2025) The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Mathematical Sciences and Applications E-Notes 13 2 65–71.
IEEE Y. Alp, “The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials”, Math. Sci. Appl. E-Notes, vol. 13, no. 2, pp. 65–71, 2025, doi: 10.36753/mathenot.1607457.
ISNAD Alp, Yasemin. “The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials”. Mathematical Sciences and Applications E-Notes 13/2 (June 2025), 65-71. https://doi.org/10.36753/mathenot.1607457.
JAMA Alp Y. The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Math. Sci. Appl. E-Notes. 2025;13:65–71.
MLA Alp, Yasemin. “The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 2, 2025, pp. 65-71, doi:10.36753/mathenot.1607457.
Vancouver Alp Y. The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Math. Sci. Appl. E-Notes. 2025;13(2):65-71.

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