On the k- Pell Quaternions and the k- Pell-Lucas Quaternions

Year 2018, Volume: 8 Issue: 1, 23 - 35, 31.03.2018

Kübra Gül

https://doi.org/10.21597/jist.407804

Cited By: 3

Abstract

The quaternions form a four-dimensional associative and non-commutative algebra over the set of

real numbers. In this paper, firstly, we give some relations for k - Pell quaternions and k - Pell-Lucas quaternions.

Then, by using Binet’s formula, we obtain their sums formulas, their the identities such as Cassini’s identity and

generating function, also derive relationships between these quaternions.

Keywords

k - Pell numbers , k - Pell-Lucas numbers , quaternions

k- Pell Kuaterniyonlar ve k- Pell-Lucas Kuaterniyonlar Üzerine

Abstract

Kuaterniyonlar, reel sayılar kümesinde dört boyutlu birleşmeli ve değişmeli olmayan bir cebir oluştururlar.

Bu makalede ilk olarak, k - Pell kuaterniyonlar ve k - Pell-Lucas kuaterniyonların bazı bağıntılarını verdik. Daha

sonra, Binet formülünü kullanarak toplam formüllerini, Cassini özdeşliği ve geren fonksiyonu gibi özdeşliklerini

elde ettik, ayrıca bu kuaterniyonların arasındaki ilişkileri türettik.

Keywords

k - Pell sayıları , k - Pell-Lucas sayıları , kuaterniyonlar

References

• Bolat C, Köse H, 2010. On the properties of k-Fibonacci numbers. Int. J. Contemp. Math. Sciences, 5(22):1097-1105.
• Catarino P, 2013. On some identities and generating functions for k-Pell numbers. Int. Journal of Math. Analysis, 7:1877-1884.
• Catarino P, 2016. The Modified Pell and the Modified k-Pell quaternions and octonions. Adv. Appl. Clifford Algebras, 26:577-590.
• Catarino P, Vasco P, 2013. Some basic properties and a two-by-two matrix involving the k-Pell numbers. Int. Journal of Math. Analysis, 7:2209-2215.
• Catarino P, Vasco P, 2013. On some identities and generating functions for k-Pell-Lucas sequence. Applied Mathematical Sciences, 7(98):4867-4873.
• Cerin Z, Gianella GM, 2007. On sums of Pell numbers. Acc. Sc. Torino-Atti Sc. Fis., 141:23-31.
• Cerin Z, Gianella GM, 2006. On sums of squares of Pell-Lucas numbers. Integers J. Comb. Number Theory, 6:1-16.
• Çimen CB, İpek A, 2016. On Pell quaternions and Pell-Lucas quaternions. Adv. Appl. Clifford Algebras, 26: 39-51.
• Everest G, 2005. An introduction to number theory. Graduate Texts in Matematics, London, 294 p.
• Falcon S, 2011. On the k-Lucas numbers. Int. J. Contemp. Math. Sciences, 6(21):1039-1050.
• Falcon S, Plaza A, 2007. On the Fibonacci k-numbers. Chaos Solitons Fractals, 32(5):1615-1624.
• Halici S, 2012. On Fibonacci quaternions. Adv. Appl. Clifford Algebras, 22:321-327.
• Horadam AF, 1963. Complex Fibonacci numbers and Fibonacci quaternions. Am. Math Quart., 70: 289-291.
• Horadam AF, 1971. Pell identities. Fibonacci Quart., 9: 245-252.
• Horadam AF, 1993. Quaternion recurrence relations. Ulam Quart., 2: 23-33.
• Iakin AL, 1981. Extended Binet forms for generalizaed quaternions of higher order. The Fib. Quaterly, 19, 410-413.
• Iyer MR, 1969. A note on Fibonacci quaternions. The Fib. Quaterly, 3:225-229.
• Iyer MR, 1969. Some results on Fibonacci quaternions. The Fib. Quaterly, 7:201-210.
• Koshy T, 2001. Fibonacci and Lucas Numbers with Applications. A Wiley-Interscience Publication, Newyork, 672 p.
• Ramirez JL, 2015. Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. An. Şt. Univ. Ovidus Constanta, 23(2): 201-2012.
• Swamy MN, 1973. On generalized Fibonacci quaternions. The Fib. Quaterly, 5:547-550.
• Szynal-Liana A, Wloch I, 2016. The Pell quaternions and the Pell octions, Adv. Appl. Clifford Algebras, 26:435-440.
• Vasco P, Catarino P, Campos H, Aires AP, Borges A, 2015. k-Pell, k- Pell-Lucas and Modified k-Pell numbers: Some identities and norms of Hankel matrices. Int. Journal of Math. Analysis, 9(1):31-37.