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

Yasemin Alp

doi:10.36753/mathenot.1607457

EN

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

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).

Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Yasemin Alp ^*
0000-0002-4146-7374
Türkiye

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 Volume: 13 Number: 2

DOI

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

IZ

https://izlik.org/JA68NY45YK

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

1.Alp Y. The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Math. Sci. Appl. E-Notes. 2025;13(2):65-71. doi:10.36753/mathenot.1607457

Chicago

Alp, Yasemin. 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.

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

[1]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, June 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 1, 2025): 65-71. https://doi.org/10.36753/mathenot.1607457.

JAMA

1.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, June 2025, pp. 65-71, doi:10.36753/mathenot.1607457.

Vancouver

1.Yasemin Alp. The Generalized Binomial Transform of the Bivariate Fibonacci and Lucas $p$-Polynomials. Math. Sci. Appl. E-Notes. 2025 Jun. 1;13(2):65-71. doi:10.36753/mathenot.1607457