Some (s, t) –Pell difference sequences

Year 2024, Volume: 13 Issue: 3, 112 - 122, 31.12.2024

Şükran Uygun , Ozan Haklıdır , Ugur Aslan

Abstract

In this study, we apply the concept of difference relation to the sequences of the (s,t)-Pell, (s,t)-Pell-Lucas, and modified (s,t)-Pell numbers. We will obtain formulas to find any term of the i th element of the (s,t)-Pell, (s,t)-Pell-Lucas, and modified (s,t)-Pell difference sequence from the initial (s,t)-Pell, (s,t)-Pell-Lucas, and modified (s,t)-Pell numbers. We will also find formulas for the sum of the elements of these new sequences as well as their generating functions, Binet formulas and some algebraic properties.

Keywords

Pell sequence, Pell-Lucas sequence, Modified Pell Sequence, Difference sequence

(s, t) –Pell Fark Dizileri

Abstract

Bu çalışmada, fark bağıntısı kavramını (s,t)-Pell, (s,t)-Pell-Lucas ve modifiye (s,t)-Pell sayı dizilerine uygulayacağız. (s,t)-Pell, (s,t)-Pell-Lucas ve modifiye (s,t)-Pell dizilerinin i. fark dizilerinin herhangi bir elemanını (s,t)-Pell, (s,t)-Pell-Lucas ve modifiye (s,t)-Pell sayılarını kullanarak bulmak için formüller oluşturacağız. Ayrıca bu yeni fark dizilerinin elemanlarının toplam formülleri, üreteç fonksiyonları, Binet formülleri ve bazı cebirsel özelliklerini inceleyeceğiz.

Keywords

Fark dizisi, Pell dizisi, Pell-Lucas dizisi

References

• [1] Catarino, P., 2013. On some identities and generating functions for k-Pell numbers, Int. J. Math. Anal. (Ruse), 7(38), 1877-1884.
• [2] Catarino, P., 2013. A note involving two-by-two matrices of the k-Pell and k-Pell-Lucas sequences, Int. Math. Forum, 8(32), 1561-1568.
• [3] Catarino, P., Vasco, P., 2013. On Some Identities and Generating Functions for k-Pell-Lucas Sequence, Appl. Math. Sci., 7(98), 4867-4873.
• [4] Catarino, P., Vasco, P., 2013. Modified k-Pell Sequence: Some Identities and Ordinary Generating Function, Appl. Math. Sci., 7(121), 6031-6037.
• [5] Vasco, P ., Catarino, P., 2014. Sums and products involving terms of k-Pell, k-Pell-Lucas and Modified k-Pell sequences”, JP J. Algebra, Number Theory Appl., 32(2), 87-98.
• [6] Petroudi SHJ, Pirouz B. 2015. On some properties of (k, h)-Pell sequence and (k, h)-Pell-Lucas sequence, International Journal of Advances in Applied Mathmatics and Mechanics. 3(1), 98-101.
• [7] Gulec, H. H., Taskara, N., 2012. On the (s, t)-Pell and (s, t)-Pell.Lucas sequences and their matrix representations, Applied Mathematics Letters, 25(10), 1554-1559.
• [8] Uygun, Ş., Açar, Z. S., 2023. Notes on (s, t)-Pell and (s, t)-Pell Lucas matrix sequences, Asian Journal of Mathematics and Physics, 7 Article ID 1
• [9] Spelina, LT Wloch. I. On Generalized Pell and Pell-Lucas Numbers, Iran J. Sci. Technol Sci. Available: https://doi.org/10.1007/s40995-019-00757-7
• [10] Koshy, T., 2014. Pell and Pell-Lucas numbers with applications. Springer, Berlin.
• [11] Falcon, S., 2016. The k-Fibonacci difference sequences, Chaos, Solitons Fractals, 87, 153-157.
• [12] Catarino, P., 2017. On some Pell difference sequences, MAYFEB Journal of Mathematics , 4, 73-84.
• [13] Catarino, P., 2018. On Jacobsthal Dıfference Sequences, Acta Math. Univ. Comenianae, LXXXVII, 267-276.
• [14] https://en.wikipedia.org/wiki/Recurrence relation.