On Gaussian Balancing and Gaussian Cobalancing Quaternions

In this paper, we present and study new kinds of sequence of quaternion numbers called as Gaussian Balancing and Gaussian Cobalancing Quaternions involving some interesting results, Binet formula and generating functions. We show matrix representations for these quaternions. Thus, we have carried the quaternions to the complex space.

Keywords:

Balancing numbers, cobalancing numbers, gaussian balancing numbers, gaussian cobalancing numbers, gaussian balancing quaternions gaussian cobalancing quaternions, matrix representations,

PDF

___

[1] Behera, A., Panda, G.K., On the square roots of triangular numbers, The Fibonacci Quarterly, 37(2)(1999), 98-105.
[2] Berczes, A., Liptai, K., Pink, I., On generalized balancing sequences, The Fibonacci Ouarterly,48(2)(2010), 121-128.
[3] Çimen, B. C., Ipek, A., On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Cliord Algebras 26(1)(2016), 39-51.
[4] Gürlebeck, K., Sprössig, W., Quaternionic and Cliord Calculus for Physicists and Engineers, Wiley, New York, 1997.
[5] Halici, S., On Fibonacci quaternions, Adv. Appl. Cliord Algebras, 22(2012), 321-327.
[6] Hamilton, W. R., Elements of Quaternions Longmans, Green and Co., London, 1866.
[7] Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, American Math. Monthly, 70(1963), 289-291.
[8] Horadam, A. F., Quaternion recurrence relations, Ulam Quarterly, 2(1993), 23-33.
[9] Iyer, M. R., A Note on Fibonacci quaternions, The Fibonacci Quarterly, 3(1969), 225-229.
[10] Keçilioğlu, O., Akkus, I., The Fibonacci octanions, Adv. Appl. Cliord Algebras, 25(2015), 151-158.
[11] Olajos, P., Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37(2010), 125- 138.
[12] Panda, G.K., Some fascinating properties of balancing numbers, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, Cong. Numer, 194(2009), 185-189.
[13] Panda, G.K., Sequence balancing and cobalancing numbers, The Fibonacci Quarterly, 45(2007), 265-271.
[14] Panda, G.K., Ray, P.K., Cobalancing numbers and cobalancer, Int. J. Math. Sci., 8(2005), 1189-1200.
[15] Panda, G.K., Ray, P.K., Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6(2011), 41-72.
[16] Ray, P.K., Panda, G. K., Balancing and Cobalancing Numbers, PhD Thesis, Department of Mathematics, National Institute of Technology, Rourkela, India, 2009.
[17] Ray, P.K., Certain Matrices associated with balancing and Lucas-balancing numbers, Matematika, 28(1)(2012), 15-22.
[18] Ray, P.K., Factorizations of the negatively subscripted balancing and Lucas-balancing numbers, Bol. Soc. Paran. Mat.,31(2)(2013), 161-173.
[19] Sevgi, E., Taşçı, D., Bi-periodic balancing quaternions, Turkish Journal of Mathematics and Computer Science , 12(2)(2020), 68-75.
[20] Szynal-Liana, A., Wloch, I., A note on jacobsthal quaternions, Adv. Appl. Clifford Algebras, 26(2016), 441-447.
[21] Tasci, D., Gaussian balancing numbers and gaussian Lucas-balancing numbers, Journal of Science and Arts, 3(44)(2018), 661-666.
[22] Ward, J.P., Quaternions and Cayley Numbers: Algebra and Applications, Kluwer Academic Publishers, London, 1997.
[23] Yilmaz, M., Asci, M., On Gaussian Balancing and Gaussian Cobalancing Numbers, Master Thesis, Department of Mathematics, Pamukkale University, Institute of Science, Denizli, Turkey, 2017.

Turkish Journal of Mathematics and Computer Science-Cover

ISSN: 2148-1830
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2013
Yayıncı: MATEMATİKÇİLER DERNEĞİ

Arşiv