Some new families of generalized $k$-Leonardo and Gaussian Leonardo Numbers

Document Type : Original paper


Department of Mathematics, Central University of Jharkhand, India, 835205


In this paper, we introduce a new family of the generalized $k$-Leonardo numbers and study their properties. We investigate the Gaussian Leonardo numbers and associated new families of these Gaussian forms. We obtain combinatorial identities like Binet formula, Cassini's identity, partial sum, etc. in the closed form. Moreover, we give various generating and exponential generating functions.


Main Subjects

[1] Y. Alp and E.G. Koçer, Some properties of Leonardo numbers, Konuralp J. Math. 9 (2021), no. 1, 183–189.
[2] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. 89 (2019), no. 1, 75–86.
[3] P. Catarino, P. Vasco, H. Campos, A.P. Aires, and A. Borges, New families of Jacobsthal and Jacobsthal-Lucas numbers, Algebra Discrete Math. 20 (2015), no. 1, 40–54.
4] M. El-Mikkawy and T. Sogabe, A new family of $k$-Fibonacci numbers, Appl. Math. Comput. 215 (2010), no. 12, 4456–4461.
[5] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2019.
[6] K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers, Integer 22 (2022), Article number A48.
[7] M. Kumari, J. Tanti, and K. Prasad, On some new families of $k$-Mersenne and generalized $k$-Gaussian Mersenne numbers and their polynomials, Contrib. Discrete Math. 18 (2023), no. 2, 244–260.
[8] E. Özkan, İ. Altun, and A. Göçer, On relationship among a new family of $k$-Fibonacci, $k$-Lucas numbers, Fibonacci and Lucas numbers, Chiang Mai J. Sci. 44 (2017), no. 4, 1744–1750.
[9] E. Özkan and M. Taştan, On a new family of Gauss $k$-Lucas numbers and their polynomials, Asian-Eur. J. Math. 14 (2021), no. 6, Article ID: 2150101.
[10] K. Prasad, H. Mahato, and M. Kumari, On the generalized $k$-Horadam-like sequences, Algebra, Analysis, and Associated Topics, Springer, 2022, pp. 11–26.
[11] P.K. Ray, On the properties of $k$-balancing and $k$-Lucas-balancing numbers, Acta Comment. Univ. Tartu. Math. 21 (2017), no. 2, 259–274.
[12] P.K. Ray, On the properties of $k$-balancing numbers, Ain Shams Engineering J. 9 (2018), no. 3, 395–402.
[13] A.G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math. 25 (2019), no. 3, 97–101.–101
[14] A.G. Shannon and Ö. Deveci, A note on generalized and extended Leonardo sequences, Notes Number Theory Discrete Math. 28 (2022), no. 1, 109–114.–114
[15] M. Shattuck, Combinatorial proofs of identities for the generalized Leonardo numbers, Notes Number Theory Discrete Math. 28 (2022), no. 4, 778–790.
[16] Y. Soykan, Generalized Leonardo numbers, J. Progressive Res. Math. 18 (2021), no. 4, 58–84.
[17] Y. Soykan, Special cases of generalized Leonardo numbers: Modified $p$-Leonardo, $p$-Leonardo-Lucas and $p$-Leonardo numbers, Earthline J. Math. Sci. 11 (2023), no. 2, 317–342.
[18] M. Taştan, N. Yılmaz, and E. Özkan, A new family of Gauss $(k, t)$-Horadam numbers, Asian-Eur. J. Math. 15 (2022), no. 12, Article ID: 2250225.
[19] Y. Yazlik and N. Taskara, A note on generalized k-Horadam sequence, Comput. Math. Appl. 63 (2012), no. 1, 36–41.
[20] N. Yilmaz, M. Taştan, and E. Özkan, A new family of Horadam numbers, Electron. J. Math. Anal. Appl. 10 (2022), no. 1, 64–70