Hybrid numbers with balancing and Lucas balancing hybrid number coefficients

Year 2025, Volume: 74 Issue: 3, 395 - 402, 23.09.2025

Murat Turan , Sıddıka Özkaldı Karakuş

https://doi.org/10.31801/cfsuasmas.1561936

https://izlik.org/JA23HF32ZE

Abstract

In this note, we made a connection between hybrid numbers and hybrid balancing, and hybrid Lucas balancing number. Initially, we obtained for these new number recurrence relation, some important relations among new numbers, and Binet's like formula. Afterwards, using Binet's like formula, we obtained some identities such as D'ocagne's identity, Honsberger identity, Catalan identity, and Cassini identity.

Keywords

Hybrid numbers , balancing numbers , Lucas numbers

References

• Alp, Y., Koçer, E. G., Hybrid Leonardo numbers, Chaos, Solitons and Fractals, 150 (2021), 111128.
• Behera, A., Panda, G. K., On the square roots of triangular numbers, Fibonacci Quart., 37(2) (1999), 98-105.
• Berczes, A., Liptai, K., Pink, I., On generalized balancing numbers, Fibonacci Quart., 48(2) (2010), 121-128.
• Brod, D., Szynal-Liana, A., Wloch, I., Two generalizations of dual-hyperbolic balancing numbers, Symmetry, 12 (2020), 1866.
• Catarino, P., Campos, H., Vasco, P., On some identities for balancing and cobalancing numbers, Annales Math. et Inform., 45 (2015), 11-24.
• Davala, R. K., Panda, G. K., On sum and ratio formulas for balancing numbers, Journal of the Indian Math. Soc., 82(1-2) (2015), 23-32.
• Erkan, E, Dağdeviren, A., k-Fibonacci and k-Lucas hybrid numbers, Tamap J. of Math. and Sta., (2021), Article id: 125.
• Gozeri, G. K., Ozkoc, A., Tekcan, A., Some algebraic relations on balancing numbers, Utilitas Mathematica, 103 (2017), 217-236.
• Keskin, R., Karaatlı, O., Some new properties of balancing numbers and square triangular numbers, Journal of Integer Seq., 15 (2012), Article 12.1.4.
• Kızılateş, C., A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos, Solitons and Fractals, 130 (2020), 109449.
• Liptai, K., Fibonacci balancing numbers, Fibonacci Quart., 42(4) (2004), 330-340.
• Liptai, K., Lucas balancing numbers, Acta Math. Univ. Ostrav., 14(1) (2006), 43-47.
• Liptai, K., Luca, F., Pinter, A., Szalay, L., Generalized balancing numbers, Indigationes Math., 20(1) (2009), 87-100.
• Olajos, P., Properties of balancing, cobalancing and generalized balancing numbers, Annales Math. Inform., 37 (2010), 125-138.
• Özdemir, M., Introduction to hybrid numbers, Adv. Appl. Clifford Algebras, 28(11) (2018).
• Panda, G. K., Ray, P. K., Cobalancing numbers and cobalancers, Int. J. Math. and Math. Sci., 8 (2005), 1189-1200.
• Panda, G. K., Ray, P. K., Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. of the Inst. of Math. Academia Sinica (New Series), 6(1) (2011), 41-72.
• Panda, G. K., Sequence balancing and cobalancing numbers, Fibonacci Quart., 45(3) (2007), 265-271.
• Panda, G. K., Some fascinating properties of balancing numbers, Proc. of the 11th Int. Conf. on Fibonacci Numbers and Their Appl., 194 (2009), 185-189.
• Polatlı, E., Hybrid numbers with Fibonacci and Lucas hybrid number coefficients, Univ. J. Math. Appl., 6(3) (2023), 106–113. https://doi.org/10.32323/ujma.1339603.
• Ray, P. K., Balancing and Cobalancing Numbers, Ph. D. Thesis (2009), National Institute of Tech., Rourkela, India.
• Ray, P. K., Certain matrices associated with balancing and Lucas-balancing numbers, Matematika, 28(1) (2012), 15-22.
• Ray, P. K., Factorizations of the negatively subscripted balancing and Lucas-balancing numbers, Bol. Soc. Paran. Mat., 31(2) (2013), 161-173.
• Ray, P. K., On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3(1) (2014), 1-6.
• Ray, P. K., Sahu, J., Generating functions for certain balancing and Lucas-balancing numbers, Palestine J. of Math., 5(2) (2016), 122-129.
• Szynal-Liana, A., The Horadam hybrid numbers, Discussiones Math., 38 (2018), 91-98.
• Szynal-Liana, A., Wloch, I., On Jacobsthal and Jacobsthal-Lucas hybrid numbers, Annales Math. Silesianae, Sciendo, 33(1) (2019), 276-283.
• Şentürk, T. D., Bilgici, G., Daşdemir, A., Ünal, Z., A study on Horadam hybrid numbers, Turk. J. of Math., 44 (2020), 1212-1221.
• Uysal, M., Özkan, E., Balancing and Lucas-balancing hybrid numbers and some identities, Journal of Information & Optimization Sciences, 45(5) (2024), 1293-1304.