Hybrinomials Related to Hyper-Fibonacci and Hyper-Lucas Numbers

Year 2023, Volume: 8 Issue: 1, 1 - 13, 30.04.2023

Efruz Özlem Mersin

https://doi.org/10.30931/jetas.1196595

Abstract

Hybrid number system is a generalization of complex, hyperbolic and dual numbers. Hybrid numbers and hybrid polynomials have been the subject of much research in recent years. In this paper, hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers are defined. Then some algebraic properties of newly defined hybrinomials are examined such as the recurrence relations and summation formulas. Also, the relation between hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers is given. Additionally, hybrid hyper-Fibonacci and hybrid hyper-Lucas numbers are defined by using the hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers.

Keywords

Hyper-Fibonacci numbers, hyper-Lucas numbers, polynomials, hybrinomials

References

• Bahşi M., Mező I., Solak S., "A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers", Annales Mathematicae et Informaticae 43 (2014): 19-27.
• Bilgici G., "New generalizations of Fibonacci and Lucas sequences", Applied Mathematical Sciences 8(29) (2014) : 1429-1437.
• Bilgici G., "Two generalizations of Lucas sequence", Applied Mathematics and Computation 245 (2014): 526-538.
• Catarino P., "On k-Pell hybrid numbers", Journal. of Discrete Mathematical Sciences and Cryptography 22 (2019):.83-89. Doi:10.1080/09720529.2019.1569822
• Cerda-Morales G., "Investigation of generalized hybrid Fibonacci numbers and their Properties*", Applied Mathematics E-Notes 21 (2021) : 110-118.
• Dil A. and Mező I., "A symmetric algorithm hyperharmonic and Fibonacci numbers". Applied Mathematics and Computation, 206 (2008) : 942-951.
• Falcon S. and Plaza A., "On the Fibonacci k-numbers", Chaos, Solitons and Fractals 32(5) (2007) : 1615-1624.
• Gupta V.K., Panwar Y.K. and Sikhwal O., "Generalized Fibonacci sequences", Theoretical Mathematics and Applications 2(2) (2012) : 115-124.
• Horzum T. and Kocer E.G., "On some properties of Horadam polynomials", International Mathematical Forum 4(25) (2009) : 1243-1252.
• Kızılateş C., "A note on Horadam hybrinomials", Fundamental Journal of Mathematics and Applications 5(1) (2022) : 1-9. Doi:10.33401/fujma.993546
• Kızılateş C., "A new generalization of Fibonacci hybrid and Lucas hybrid numbers", Chaos, Solitons and Fractals 130 (2020) : 109449.
• Kocer E.G., Tuglu N. and Stakhov A., "On the m-extension of the Fibonacci and Lucas p-numbers", Chaos, Solitons and Fractals 40(4) (2009) :1890-1906.
• Koshy T., "Fibonacci and Lucas numbers with applications Pure and Applied Mathematics", A Wiley-Interscience Series of Texts, Monographs, and Tracts, New York: Wiley, (2001).
• Köme C., Yazlık Y. and Madhusudanan V., "A new generalization of Fibonacci and Lucas p-numbers", Journal of Computational Analysis and Applications 25(4) (2018) : 657-669.
• Lengyel T., "The order of the Fibonacci and Lucas numbers", Fibonacci Quarterly 33(3) (1995) : 234-239.
• Mersin E.Ö., "Hyper-Fibonacci and hyper-Lucas polynomials", Conference Proceeding of 5th International E-Conference on Mathematical Advances and Applications (ICOMAA-2022), Yildiz Technical University, Istanbul, Turkey, (2022).
• Mersin E.Ö. and Bahşi M., "Hyper-Fibonacci and hyper-Lucas hybrinomials", Konuralp Journal of Mathematics 10(2) (2022) : 293-300.
• Muskat J.B., "Generalized Fibonacci and Lucas sequences and rootfinding methods", Mathematics and Computation, 61(203) (1993) : 365-372.
• Özdemir M., "Introduction to hybrid numbers", Advances in Applied Clifford Algebras, 28(11) (2018). Doi:10.1007/s00006-018-0833-3
• Özkan E. and Uysal M., "Mersenne-Lucas Hybrid numbers", Mathematica Montisnigri 52 (2021) : 17-29.
• Szynal-Liana A., "The Horadam hybrid numbers", Discussiones Mathematicae General Algebra and Applications 38 (2018) : 91-98. Doi: 10.7151/dmgaa.1287
• Szynal-Liana A. and Wloch I., "On Jacobsthal and Jacobsthal-Lucas Hybrid numbers", Annales Mathematicae Silesianae 33 (2019) : 276-283. Doi:10.2478/amsil-2018-0009
• Szynal-Liana A. and Wloch I., "Introduction to Fibonacci and Lucas hybrinomials". Complex Variables and Elliptic Equations, 65(10) (2020) : 1736-1747.
• Şentürk T.D., Bilgici G., Daşdemir A. and Ünal Z., "A study on Horadam hybrid numbers", Turkish Journal of Mathematics 44(4) (2020) : 1212-1221.
• Tan E. and Ait-Amrane N.R., "On a new generalization of Fibonacci hybrid numbers", Indian Journal of Pure and Applied Mathematics (2022) : 1-11.
• Tasci D. and Firengiz M.C., "Incomplete Fibonacci and Lucas p-numbers", Mathematical and Computer Modelling 52(9-10) (2010) : 1763-1770.
• Yılmaz N., "More identities on Fibonacci and Lucas hybrid numbers", Notes on Number Theory and Discrete Mathematics 27(2) (2021) : 159-167. Doi:10.7546/nntdm.2021.27.2.159-167
• Yayenie O., "A note on generalized Fibonacci sequences", Applied Mathematics and Computation 217(12)(2011) : 5603-5611.