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

Year 2025, Volume: 13 Issue: 2, 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).