Catalan Transform of The ? −Lucas Numbers

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

Catalan Transform of The ? −Lucas Numbers

In this study, the ???,? description of Catalan transformation of ? −Lucas ??,? sequences was given. The ???,? generating function of Catalan transformation of ? −Lucas ??,? sequences was obtained. And also, ???,? transformation was written as the multiplying of Catalan matris C which is the lower triangular matris, and the ?? matris of ? ? 1 type. Determinants of matrices which were formed with ???,? by using Hankel transform were calculated.

PDF

___

Barry, P. 2005. ‘’A Catalan transform and related transformations on integer sequences’’. J.Integer Seq., 8(4), 1-24.
Cvetkovi´c, A., Rajkovi´c, R., and Ivkovi´c, M. 2002. ‘’Catalan numbers, and Hankel transform, and Fibonacci numbers’’ J. Integer Seq. 5(1), 1-8.
Layman, J. W. 2001. ‘’The Hankel transform and some of its properties’’. J. Integer Seq. 4(1), 1-11.
Rajkovi´c, P. M., Petkovi´c, M. D. and Barry, P. 2007. ‘’The Hankel transform of the sum of consecutive generalized Catalan numbers. Integral Transforms Spec. Funct’’. 18(4), 285–296.
Horadam, A. F. 1961. ‘’A generalized Fibonacci sequence’’. Amer. Math. Monthly, 68 (5), 455–459.
Bruhn, H. Gellert, L. Günther, J. 2015. ‘’Jacobsthal numbers in generalized Petersen graphs’’ Electronic Notes in Discrete Mathematics, 49, 465-472.
Anuradha, N. 2008. ‘’Number of points on certain hyperelliptic curves defined over finite fields’’ Finite Fields and Their Applications, 14, 314-328.
Falcon, S. (2013). ‘’Catalan Transform of the k-Fibonacci sequence’’. Commun. Korean Math. Soc. 28(4), 827–832.
Özkan, E., Taştan, M., Aydoğdu, A. 2018. ‘’2-Fibonacci Polynomials in the Family of Fibonacci Numbers’’, Notes on Number Theory and Discrete Mathematics, 24, 47-55.
Koshy, T. 2001. ‘’Fibonacci and Lucas Numbers with Applications’’. New York, NY: Wiley-Interscience Publication.
Falc´on, S., and Plaza, A. 2007. ‘’On the Fibonacci k-numbers’’, Chaos Solitons Fractals 32(5), 1615–1624.
Sloane, N. J. A., 2007. The on-line encyclopedia of integer sequences, www.research.att.com/˜njas/sequences/ http://en.wikipedia.org/wiki/Catalan number.
Boussayoud, A., Kerada, M. and , Harrouche, N. 2017. ‘’On the k -Lucas Numbers and Lucas Polynomials’’, Turkish Journal of Analysis and Number Theory, 5(4), 121-125.
Özkan E., Taştan M. 2019."On Gauss Fibonacci Polynomials, Gauss Lucas Polynomials and Their Applications ", Communications in Algebra, https://doi.org/10.1080/00927872.2019.1670 193
Özkan E., Altun İ. 2019. "Generalized Lucas Polynomials And Relationships Between The Fıbonacci Polynomials And Lucas Polynomials ", Communications in Algebra, 47, 10-12.
Özkan E., Altun İ., Göçer A. 2017, "On Relationship Among A New Family of kFibonacci, k-Lucas Numbers, Fibonacci And Lucas Numbers", Chiang Mai Journal of Scıence, 44, 1744-1750.
Özkan E. 2014. "Truncated Lucas sequence and its period", Applied Mathematics and Computation, 232, 285-291.
Taştan, M., Özkan, E. 2020. ‘’Catalan Transform of The k−Jacobsthal Sequence’’, Electronic Journal of Mathematical Analysis and Applications, 8(2), 70-74.
Özkan, E., Taştan, M., Aydoğdu, A. 2019. ‘’3-Fibonacci Polynomials in The Family of Fibonacci Numbers’’, Erzincan University Journal of the Institute of Science and Technology, 12(2), 926-933.
Özkan, E., Taştan, M. 2019. ‘’? −Fibonacci Polynomials in The Family of Fibonacci Numbers’’, Research & Reviews: Discrete Mathematical Structure, 6(3),19-22.