Değiştirilmiş Pell ve Pell Lucas Dizilerinin Özellikleri

Yıl 2019, Cilt: 9 Sayı: 3, 1646 - 1656, 01.09.2019

Fikri Köken

https://doi.org/10.21597/jist.538046

Cited By: 1

Öz

Değiştirilmiş Pell ve Pell Lucas dizileri, Pell ve Pell Lucas sayıları değiştirilerek tanımlanır, bu dizilerin, Pell ve Pell Lucas dizileriyle benzer özelliklere sahip oldukları görülmektedir. Bu nedenle, değiştirilmiş dizilerin bazı indirgeme özelliklerini incelenir. Ayrıca, değiştirilmiş dizilerin en büyük ortak bölenleri (yani, EBOB) dizileri araştırılır ve EBOB dizilerinin, Pell ve Pell Lucas dizilerinin alt dizileri olduğu görülür. Bu nedenle, GCD dizilerinin Binet formülünü, Cassini, Catalan ve Docagne’nin eşitlikleri elde edilir.

Anahtar Kelimeler

Değiştirilmiş Pell ve Pell Lucas sayıları, EBOB Dizileri, İndirgeme bağıntıları

The Properties of the Altered Pell and Pell Lucas Sequences

Öz

The altered Pell and Pell Lucas sequences are defined by altering the Pell and Pell Lucas numbers, it is seen that they have similar properties to usual the Pell and Pell Lucas sequences. Thus, we study some recursive properties of the altered sequences. Further, the greatest common divisors (i.e. GCD) sequences of the altered sequences are investigated, and it is seen that the GCD sequences are subsequences of the Pell and Pell Lucas sequences. Therefore, we obtain Binet formula, the Cassini, Catalan and D’ocagne’s identities of the GCD sequences.

Anahtar Kelimeler

Altered Pell and Pell Lucas numbers, GCD Sequences, Recurrence relations

Kaynakça

• Bicknell N, 1975. A primer on the Pell sequence and related sequence, Fibonacci Quart. 13(4), 345–349.
• Chen KW, 2011. Greatest common divisors in shifted Fibonacci sequences. J. Integer. Seq. 14, 11. 4–7.
• Dudley U, Tucker B, 1971. Greatest common divisors in altered Fibonacci sequences. Fibonacci Quarterly, 9: 89–91.
• Gül K, 2018. On the k Pell Quaternions and the k Pell Lucas Quaternions. Iğdır University Journal of the Institute of Science and Technology, 8(1): 23–35.
• Halıcı S, Oz S, 2016. On Some Gaussian Pell And Pell Lucas Numbers. Ordu University Journal of Science and Technology, 6(1): 8–18.
• Halıcı S, Oz S, 2018. On Gaussian Pell Polynomials and Their Some Properties. Palestine Journal of Mathematics 7(1), 251–256.
• Horadam AF, 1971. Pell identities, Fibonacci Quart. 9(3), 245–263.
• Horadam AF, Mahon JM, 1985. Pell and Pell-Lucas Polynomials, Fibonacci Quart. 23(1), 7–20.
• Horadam AF, 1994. Applications of Modified Pell Numbers to Representations. Ulam Quart., 3, 34–53.
• Karakas M, Karatas A.M, 2017. New Banach sequence spaces that is defined by the aid of Lucas numbers. Iğdır University Journal of the Institute of Science and Technology, 7(4): 103–111.
• Koken F, Arslan, S, 2018. GCD Properties of the Altered Pell And Pell Lucas Numbers. IOSR Journal of Mathematics, 14(5): 82–89.
• Koshy T, 2001. Fibonacci and Lucas Numbers with Applications. A Wiley-Interscience Publication, Newyork, 672 p.
• Koshy T, 2014. Pell and Pell Lucas Numbers with Applications. Springer, Berlin, 444 p.
• Mahon JM, Horadam AF, 1986. Matrix and Other Summation Techniques for Pell Polynomials. Fibonacci Quart. 24(4), 290–308.
• McDaniel WL, 1991. The G.C.D. in Lucas sequences and Lehmer number sequences, Fibonacci Quart. 29, 24–29.
• Tasyurdu Y, Cobanoğlu N, Dilmen Z, 2016. On the a New Family of k-Fibonacci Numbers. Erzincan University Journal of Science and Technology, 9(1): 95–101.