Van Der Laan Hibrit Dizileri Üzerine

Year 2021, , 370 - 381, 31.08.2021

Seyyed Hossein Jafari Petroudi , Maryam Pirouz

https://doi.org/10.18185/erzifbed.778102

Abstract

Bu makalede Van Der Laan hibrit dizisi tanıtıldı. Bu dizi ile ilişkili Binet benzeri formül, kısmi toplam ve üreteç fonksiyonu elde edildi. Van Der Laan hibrit dizisinin bazı ilginç özellikleri verildi. Son olarak, Van Der Laan hibrit dizisini içeren bir sirkülant matrisin öz değerleri ve determinantı sunuldu.

Keywords

Van Der Laan dizisi, hibrit sayılar, kısmi toplam, üreteç fonksiyonu, determinant., Van Der Laan dizisi, hibrit sayılar, kısmi toplam, üreteç fonksiyonu, determinant.

Thanks

Thanks the referees for their significant statements about the paper.

on the Van Der Laan hybrid sequence

Abstract

Many authors studied special recursion sequences such as Pell sequence, Pell Lucas sequence, Padovan and Perrin sequences, Jacobsthal sequence. They established new results about these sequences. Ozdemir (2018) introduced the hybrid numbers as a generalization of complex hyperbolic and dual numbers. The set H of hybrid numbers Z is of the form
Z=a+bi+cϵ+dh,
Where a,b,c∈R and i,ϵ,h are operators such that
i^2=-1 ,ϵ^2=0,ih=-hi= ϵ+i.
For more results about the hybrid number we refer to (Ozdemir, 2018). The conjugate of hybrid number Z is defined by
¯Z=¯(a+bi+cϵ+dh)=a-bi-cϵ-dh .

Liana and Wloch (2019) introduced the Jacobsthal and Jacobsthal Lucas hybrid numbers and investigated some of their properties. In this paper we introduce the Van Der Laan hybrid sequence. We obtain Binet-like formula of this sequence. Then we represent the partial sum and generating function of this sequence. We study some properties of this sequence. Finally we find the eigenvalues and determinant of particular circulant matrix involving van Der Laan hybrid sequence.

Keywords

Van der Laan sequence, hybrid numbers, partial sum, generating function, determinant

References

• Coskun, A. and Taskara, N. 2018. “On the some properties of circulant matrix with third order linear recurrent sequence”, Mathematical Sciences and Applications E-Notes, 6(1), 12 – 18.
• Faisant, A. 2019. “On the Padovan sequence”, arXiv, arXiv:1905.07702.
• Godase, A.D., Dhakne, M.B. 2014. “On the properties of k-Fibonacci and k-Lucas numbers”, International Journal of Advances in Applied Mathematics and Mechanics, 2(1), 100–106.
• He, T., Liao, J. H. and Shiue, P. J. 2018. “Matrix Representation of Recursive Sequences of Order 3 and Its Applications”, Journal of Mathematical Research with Applications, 38(3), 221–235.
• Kaygisiz, K. and Sahin, A. 2011. “k-sequences of Generalized Van der Laan and Generalized Perrin Polynomials”, https://www.researchgate.net/publication/51956495, 1-20.
• Liana, A.S. and Wloch, I. 2019. “Jacobsthal and Jacobsthal Lucas Hybrid numbers”, Annales Mathematiccae Silesianae, 33, 276-283.
• Morales, G.C. 2019. “New identities for Padovan sequences”, arXiv:1904.05492. Özdemir, M. 2018. “Introduction to hybrid numbers”, Advances in Applied Clifford Algebras, 28(1), 11.
• Petroudi, S.H.J. and Pirouz, B. 2015. “On some properties of (k,h)-Pell sequence and (k,h)-Pell-Lucass sequence”, International Journal of Advances in Applied Mathematics and Mechanics, 3(1), 98–101.
• Petroudi, S.H.J. and Pirouz, M. 2016. “On special circulant matrices with (k,h)-Jacobsthal sequence and (k,h)-Jacobsthal-like sequence”, Int. J. Mathematics and scientific computation, 6(1), 44-47.
• Shanon, A.G., Horadam, A. F. and Anderson, P. G. 2006. “Properties of Cordonnier, Perrin and Van Der Laan numbers”, International Journal of Mathematical Education in Science and Technology, 37(7), 825-831.
• Yilmaz, F. and Bozkurt, D. 2011. “Hessenberg matrices and the Pell and Perrin numbers”, Jounal of Number Theory, 131(8), 1390–1396.