Erzincan Üniversitesi Erzincan University Fen Bilimleri Enstitüsü Dergisi Journal of Science and Technology 2020, 13(ÖZEL SAYI-I), 145-149 2020, 13(SPECIAL ISSUE-I), 145-149 ISSN: 1307-9085, e-ISSN: 2149-4584 DOI: 10.18185/erzifbed.638488

Catalan Transform of The 풌 −Lucas Numbers

Engin ÖZKAN1 , Merve TAŞTAN2* Olcay GÜNGÖR2

1Erzincan Binali Yıldırım University, Faculty of Arts and Sciences, Departments of Mathematics , 24000, Erzincan, Turkey

2Erzincan Binali Yıldırım University, Graduate School of Natural and Applied Sciences, Department of Mathematics, 24000, Erzincan, Turkey

Geliş / Received: 25/10/2019, Kabul / Accepted: 21/02/2020

Öz

Bu çalışmada, 푘–Lucas dizisinin 퐿푘,푛 Catalan dönüşümünün 퐶퐿푘,푛 tanımı verildi. 푘–Lucas dizisinin 퐿푘,푛 Catalan dönüşümünün 퐶퐿푘,푛 geren fonksiyonu elde edildi. Ayrıca, 퐶퐿푘,푛 dönüşümü, alt üçgen matris olan Catalan matrisi 퐶 ile 푛 × 1 tipindeki 퐿푘 matrisinin çarpımı olarak yazıldı. Hankel fonksiyonu kullanılarak 퐶퐿푘,푛 ler ile oluşturulan matrislerin determinantları hesaplandı.

Anahtar Kelimeler: 푘 −Lucas dizisi, 푘 −Fibonacci dizisi, Catalan dönüşümü, Hankel dönüşümü.

Catalan Transform of The 풌 −Lucas Numbers Abstract

In this study, the 퐶퐿푘,푛 description of Catalan transformation of 푘 −Lucas 퐿푘,푛 was given. The

퐶퐿푘,푛 of Catalan transformation of 푘 −Lucas 퐿푘,푛 sequences was obtained. And also,

퐶퐿푘,푛 transformation was written as the multiplying of Catalan matris C which is the lower triangular matris, and the 퐿푘 matris of 푛 푥 1 type. of matrices which were formed with 퐶퐿푘,푛 by using Hankel transform were calculated.

Keywords: 푘 −Lucas , 푘 −Fibonacci sequence, Catalan Transform, Hankel Transform.

1. Introduction science (see e.g.[ Horadam, 1961; Bruhn, et For any 푛, it is called a generalized al., 2015; Anuradha, 2008; Özkan, et al., Fibonacci-type sequence any recurrence 2018; Falcon and Plaza, 2007; Özkan and sequence of the following form 퐺(푛 + 1) = Taştan, 2019] and the references therein) The 푎퐺(푛) + 푏퐺(푛 − 1), 퐺(0) = 푚, 퐺(1) = known Lucas numbers have some 푡 ,where 푚, 푡, 푎 and 푏 are any complex applications in many branches of numbers. There is an extensive work in the mathematics such as group theory, calculus, literature concerning Fibonacci-type applied mathematics, linear algebra, etc sequences and their applications in modern [Koshy, 2001; Özkan and Altun, 2019; 145 *Corresponding Author: [email protected]

Catalan Transform of The 푘 −Lucas Numbers

Özkan, et al., 2017; Özkan, 2014; Taştan and and Özkan, 2020; Özkan, et al., 2019; Özkan and Taştan, 2019]. 푘 − √푘2 + 4 푟 = . 2 2 There exist generalizations of the Lucas numbers. This paper is an extension of the Characteristic roots verify the properties work of Falcon [Falcon, 2013]. Falcon 2 [Falcon, 2013] gave an application of the 푟1 − 푟2 = √푘 + 4 = √∆= 훿 Catalan transform to the 푘 −Fibonacci 푟 + 푟 = 푘 sequences. In this paper, we put in for 1 2 Catalan transform to the 푘 −Lucas sequence 푟1. 푟2 = −1 and present application of the Hankel transform to the Catalan transform of the Binet’s formula for 퐿푘,푛 is 푘 −Lucas sequence. 푛 푛 퐿푘,푛 = 푟1 + 푟2 . The other section of the paper is prepared as follows. The following, we introduce some 푘 −Lucas sequence as numbered; fundamental definitions of 푘–Lucas numbers. In section 3, Catalan transform of 푘 −Lucas 퐿푘,푛+1 = 푘퐿푘,푛 + 퐿푘,푛−1 sequence is given. In section 4, generating 퐿 = 2, function of Catalan transformation of 푘,0 푘 −Lucas sequence is obtained. Finally, we give Henkel transform of the new sequence 퐿푘,1 = 푘, and its . 2 퐿푘,2 = 푘퐿푘,1 + 퐿푘,0 = 푘 + 2, 2. 풌 −Lucas numbers 2 Let k be any positive real number. Then the 퐿푘,3 = 푘퐿푘,2 + 퐿푘,1 = 푘(푘 + 2) + 1 = 푘-Lucas sequence is defined recurrently by 푘3 + 3푘, 퐿 = 푘퐿 + 퐿 for 푛 ≥ 1 푘,푛+1 푘,푛 푘,푛−1 3 2 퐿푘,4 = 푘퐿푘,3 + 퐿푘,2 = 푘(푘 + 3푘) + 푘 + 2, where 퐿푘,0 = 2 and 퐿푘,1 = 푘. We will show 4 2 the sequence with such that {퐿푘.푛}푛∈푁 from 퐿푘,4 = 푘 + 4푘 + 2, now on. 4 2 퐿푘,5 = 푘퐿푘,4 + 퐿푘,3 = 푘(푘 + 4푘 + 2) + When 푘 = 1, the known Lucas sequence is 3 optained. 푘 + 3푘,

5 3 Characteristic equation of the sequence is 퐿푘,5 = 푘 + 5푘 + 5푘,

2 푟 − 푘. 푟 − 1 = 0. 5 3 퐿푘,6 = 푘퐿푘,5 + 퐿푘,4 = 푘(푘 + 5푘 + 5푘) + Its characteristic roots are 푘4 + 4푘2 + 2,

푘 + √푘2 + 4 6 4 2 푟 = 퐿푘,6 = 푘 + 6푘 + 9푘 + 2, 1 2 146

Catalan Transform of The 푘 −Lucas Numbers

푖 6 − 푖 퐿푘,7 = 푘퐿푘,6 + 퐿푘,5 퐶퐿 = ∑3 ( ) 퐿 = 푘3 + 2푘2 + 푘,3 푖=1 6−푖 3 − 푖 푘,푖 = 푘(푘6 + 6푘4 + 9푘2 + 2) + 푘5 + 5푘3 5푘 + 4, + 5푘 푖 8 − 푖 퐶퐿 = ∑4 ( ) 퐿 = 푘4 + 3푘3 + 푘,4 푖=1 8−푖 4 − 푖 푘,푖 = 푘7 + 7푘5 + 14푘3 + 7푘. 9푘2 + 14푘 + 12,

2.1 Catalan Numbers 푖 10 − 푖 퐶퐿 = ∑5 ( ) 퐿 = 푘5 + 푘,5 푖=1 10−푖 5 − 푖 푘,푖 For 푛 ≥ 0, the 푛푡ℎ Catalan number is showed 4푘4 + 14푘3 + 30푘2 + 46푘 + 36, by [Barry, 2005]

1 (2푛)! 푖 12 − 푖 퐶 = (2푛) or 퐶 = 퐶퐿 = ∑6 ( ) 퐿 = 푘6 + 푛 푛+1 푛 푛 (푛+1)!푛! 푘,6 푖=1 12−푖 6 − 푖 푘,푖 5푘5 + 20푘4 + 53푘3 + 107푘2 + 151푘 + and its generating function is given by 114, 푐(푥) = 1−√1−4푥. 2푥 7 푖 14 − 푖 7 퐶퐿푘,7 = ∑푖=1 ( ) 퐿푘,푖 = 푘 + For some 푛, the first Catalan numbers are 14−푖 7 − 푖 6푘6 + 27푘5 + 84푘4 + 204푘3 + 378푘2 + {1, 1, 2, 5, 14, 132, 429, 1430, 4862, 16796, … } 509푘 + 372. ,from now on OEIS, as A000108 in http://en.wikipedia.org/wiki/Catalan number. We can show {퐿푘,푛} as the 푛 × 1 matrix 퐿푘 and the product of the lower triangular matrix 3. Catalan transform of the 퐶 as 풌 −Lucas sequence 퐶퐿푘,1 … 퐿푘,1 1 Following [Barry, 2005], we define the 퐶퐿 … 퐿 푘,2 1 1 푘,2 Catalan transform of the 푘 −Lucas sequence 퐶퐿푘,3 = 2 2 1 … 퐿푘,3 5 … {퐿 } as 퐶퐿푘,4 5 5 3 퐿푘,4 푘,푛 ⋱ [ ⋮ ] [⋮ ⋮ ⋮ ⋮ ] [ ⋮ ] 푛 푖 2푛−푖 퐶퐿푘,푛 = ∑푖=1 ( )퐿푘,푖 for 푛 ≥ 1 2푛−푖 푛−푖 푘 푘2 + 푘 + 2 with 퐶퐿 = 0. 3 2 푘,0 푘 + 2푘 + 5푘 + 4 푘4 + 3푘3 + 9푘2 + 14푘 + 12 We can give some members of Catalan [ ⋮ ] transform of the first 푘 −Lucas numbers. 1 … 푘 These are the polynomials in 푘: … 푘2 + 2 1 1 = 2 2 1 … 푘3 + 3푘 푖 2 − 푖 퐶퐿 = ∑1 ( ) 퐿 = 푘, 5 5 5 3 … 푘4 + 4푘2 + 2 푘,1 푖=1 2−푖 1 − 푖 푘,푖 [ ⋮ ⋮ ⋮ ⋮ ⋱ ] [ ⋮ ]

푖 4 − 푖 퐶퐿 = ∑2 ( ) 퐿 = 푘2 + 푘 + 2, where 푘,2 푖=1 4−푖 2 − 푖 푘,푖

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Catalan Transform of The 푘 −Lucas Numbers

푖−1 푎0 푎1 푎2 푎3 … 퐶푖,푗 = ∑푟=푗−1 퐶푖−1,푟. 푎1 푎2 푎3 푎4 … |푎2 푎3 푎4 푎5 …| The lower triangular matrix 퐶푛,푛−푖 is known 퐻 = 푎3 푎4 푎5 푎6 … . as the Catalan and its elements verify 푛 | . . . . . | the formula ...... (2푛 − 푖)! (푖 + 1) 퐶푛,푛−푖 = (푛 − 푖)! (푛 + 1)! The upper-left 푛 × 푛 subdeterminant of 퐻푛 is called the Hankel determinant of order 푛 of with 0 ≤ 푖 ≤ 푛. the sequence 퐴.

4. Generating function The Hankel transform of the Catalan sequence is the sequence {1, 1, 1, … } [Sloane, We know that the generating function of the 2007] and the Hankel transform of the sum 푘 −Lucas polynomials and the generating of consecutive generalized Catalan numbers function of the Catalan numbers, is the bisection of classical Fibonacci sequence [Rajkovi´c, et al.,2007]. It is very respectively, are 퐿 (푥) = 2−푘푥 and 푘 1−푘푥−푥2 interesting the study of the Catalan transform of this 푘-Lucas sequence, as we will see in 푐(푥) = 1−√1−4푥 in [1, 14]. 2푥 the sequel.

Let 퐶(푥) and 퐴(푥), respectively, be the Considering the Catalan transform of the generating function of the sequence of the 푘 −Lucas sequence of the preceding

Catalan numbers {퐶푛} and and the generating subsection, we find out: function of the sequence {푎푛}. It is proved 퐻퐶퐿 = 퐷푒푡[푘] = 푘 that 퐴(푥 ∗ 푐(푥)) is the generating function of 1 the Catalan transform of this last sequence 2 [Barry, 2005]. Consequently, we obtain that 퐻퐶퐿 = | 푘 푘 + 푘 + 2 | 2 푘2 + 푘 + 2 푘3 + 2푘2 + 5푘 + 4 the generating function of the Catalan = −4 transform of the 푘 −Lucas sequence {퐿푘,푛} is

퐻퐶퐿3 퐶퐿푘(푥) = 퐿푘(푥 ∗ 퐶(푥)) 푘 푘2 + 푘 + 2 푘3 + 2푘2 + 5푘 + 4 = | 푘2 + 푘 + 2 푘3 + 2푘2 + 5푘 + 4 푘4 + 3푘3 + 9푘2 + 14푘 + 12 | 4 − 푘 + 푘. √1 − 4푥 푘3 + 2푘2 + 5푘 + 4 푘4 + 3푘3 + 9푘2 + 14푘 + 12 푘5 + 4푘4 + 14푘3 + 30푘2 + 46푘 + 36 = . 1 + 2푥 − 푘 + (푘 + 1)√1 − 4푥 = 푘3 − 4푘2 − 8푘 − 16. 5. Hankel Transform References 퐴 = {푎 , 푎 , 푎 , … , } is a sequence of real 0 1 2 Barry, P. 2005. ‘’A Catalan transform and numbers [Cvetkovi´c, et al., 2002; Layman, related transformations on integer 2001]. The Hankel transform of the sequence sequences’’. J.Integer Seq., 8(4), 1-24. 퐴 is the sequence of determinants 퐻푛 = 퐷푒푡[푎푖+푗−2], i.e., Cvetkovi´c, A., Rajkovi´c, R., and Ivkovi´c, M. 2002. ‘’Catalan numbers, and Hankel

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Catalan Transform of The 푘 −Lucas Numbers transform, and Fibonacci numbers’’ J. http://en.wikipedia.org/wiki/Catalan number. Integer Seq. 5(1), 1-8. Boussayoud, A., Kerada, M. and , Harrouche, Layman, J. W. 2001. ‘’The Hankel transform N. 2017. ‘’On the k -Lucas Numbers and and some of its properties’’. J. Integer Seq. Lucas Polynomials’’, Turkish Journal of 4(1), 1-11. Analysis and Number Theory, 5(4), 121-125.

Rajkovi´c, P. M., Petkovi´c, M. D. and Barry, Özkan E., Taştan M. 2019."On Gauss P. 2007. ‘’The Hankel transform of the sum Fibonacci Polynomials, Gauss Lucas of consecutive generalized Catalan numbers. Polynomials and Their Applications ", Integral Transforms Spec. Funct’’. 18(4), Communications in Algebra, 285–296. https://doi.org/10.1080/00927872.2019.1670 193 Horadam, A. F. 1961. ‘’A generalized Özkan E., Altun İ. 2019. "Generalized Lucas Fibonacci sequence’’. Amer. Math. Monthly, 68 (5), 455–459. Polynomials And Relationships Between The Fıbonacci Polynomials And Lucas Bruhn, H. Gellert, L. Günther, J. 2015. Polynomials ", Communications in Algebra, ‘’Jacobsthal numbers in generalized Petersen 47, 10-12. graphs’’ Electronic Notes in Discrete Özkan E., Altun İ., Göçer A. 2017, "On Mathematics, 49, 465-472. Relationship Among A New Family of k- Anuradha, N. 2008. ‘’Number of points on Fibonacci, k-Lucas Numbers, Fibonacci And certain hyperelliptic curves defined over Lucas Numbers", Chiang Mai Journal of finite fields’’ Finite Fields and Their Scıence, 44, 1744-1750. Applications, 14, 314-328. Özkan E. 2014. "Truncated Lucas sequence and its period", Applied Mathematics and Falcon, S. (2013). ‘’Catalan Transform of the Computation, 232, 285-291. k-Fibonacci sequence’’. Commun. Korean Math. Soc. 28(4), 827–832. Taştan, M., Özkan, E. 2020. ‘’Catalan Transform of The k−Jacobsthal Sequence’’, Özkan, E., Taştan, M., Aydoğdu, A. 2018. Electronic Journal of Mathematical Analysis ‘’2-Fibonacci Polynomials in the Family of and Applications, 8(2), 70-74. Fibonacci Numbers’’, Notes on Number Theory and Discrete Mathematics, 24, 47-55. Özkan, E., Taştan, M., Aydoğdu, A. 2019. ‘’3-Fibonacci Polynomials in The Family of Koshy, T. 2001. ‘’Fibonacci and Lucas Fibonacci Numbers’’, Erzincan University Numbers with Applications’’. New York, NY: Wiley-Interscience Publication. Journal of the Institute of Science and Technology, 12(2), 926-933. Falc´on, S., and Plaza, A. 2007. ‘’On the Fibonacci k-numbers’’, Chaos Solitons Özkan, E., Taştan, M. 2019. ‘’푘 −Fibonacci Fractals 32(5), 1615–1624. Polynomials in The Family of Fibonacci Numbers’’, Research & Reviews: Discrete Sloane, N. J. A., 2007. The on-line Mathematical Structure, 6(3),19-22. encyclopedia of integer sequences, www.research.att.com/˜njas/sequences/

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