Unrestricted Pell and Pell – Lucas 2N-ons

Öz

In this study, we define unrestricted Pell and Pell – Lucas hyper-complex numbers. We choose arbitrary Pell and Pell – Lucas numbers for the coefficients of the ordered basis 〖{e〗_0,e_1,⋯,e_(N-1)} of hyper-complex 2^N-ons where N∈{0,1,2,3,4} and call these hyper-complex numbers unrestricted Pell and Pell-Lucas 2N-ons. We give generating functions and Binet formulas for these type of hyper-complex numbers. We also obtain some generalization of well – known identities such as Catalan’s, Cassini’s and d’Ocagne’s identities.

Anahtar Kelimeler

• Pell sequence
• Pell-Lucas sequence
• Quaternion
• Octonion
• Sedenion

Kaynakça

1. [1] Çağman, A., 2021. Repdigits as product of Fibonacci and Pell numbers. Turkish Journal of Science, 6(1), pp. 31-35.
2. [2] Çağman, A., 2021. An approach to Pillai's problem with the Pell sequence and the powers of 3. Miskolc Mathematical Notes, 22(2), pp. 599-610.
3. [3] Aydın F. T., Köklü K., Yüce S., 2017. Generalized dual Pell quaternions. Notes on Number Theory and Discrete Mathematics, 23(4), pp. 66-84.
4. [4] Bilgici G., Catarino P., 2018. Unrestricted Pell and Pell-Lucas quaternions. International Journal of Mathematics and Systems Science, 1(3), pp. 1-10.
5. [5] Catarino P., 2016. The modified Pell and the modified k-Pell quaternions and octonions. Advances in Applied Clifford Algebras, 26(2), pp. 577-590.
6. [6] Catarino P., 2018. Bicomplex k-Pell quaternions. Computational Methods and Function Theory, pp. 1-12.
7. [7] Catarino P., 2018. k-Pell, k-Pell–Lucas and modified k-Pell sedenions. Asian-European Journal of Mathematics, 12(2), 1950018, pp. 1-10.
8. [8] Catarino P., Vasco P., 2017. On dual - Pell quaternions and octonions. Mediterranean Journal of Mathematics, 14(2), 75.

9. [9] Çimen C.B., İpek A., 2016. On Pell quaternions and Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 26(1), pp. 39-51.
10. [10] Szynal-Liana A., Włoch I., 2016. The Pell quaternions and the Pell octonions. Advances in Applied Clifford Algebras, 26(1), pp. 435-440.
11. [11] Tokeşer Ü., Ünal Z., Bilgici G., 2017. Split Pell and Pell–Lucas quaternions. Advances in Applied Clifford Algebras, 27(2), pp. 1881-1893.
12. [12] Cawagas R.E., 2004. On the structure and zero divisors of the Cayley-Dickson sedenion algebra. Discussiones Mathematicae-General Algebra and Applications, 24(2), pp. 251-265.