Binomial Transforms of k-Narayana Sequences and Some Properties

Year 2024, Volume: 7 Issue: 3, 137 - 146, 30.09.2024

Faruk Kaplan , Arzu Özkoç Öztürk

https://doi.org/10.33401/fujma.1468536

Abstract

The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.

Keywords

$k-$Narayana Sequences, Binomial transform, Recurrence relations, Binet formulas

References

• [1] T. Koshy, Fibonacci and Lucas numbers with applications , John Wiley and Sons Inc., NY (2001). $ \href{https://doi.org/10.1002/9781118033067}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85051312326&origin=resultslist&sort=plf-f&src=s&sid=ee65ee18b37912212bc7c21b14ac96e3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fibonacci+and+Lucas+Numbers+with+Applications%22%29&sl=62&sessionSearchId=ee65ee18b37912212bc7c21b14ac96e3&relpos=0}{\mbox{[Scopus]}} $
• [2] S. Falcon and A. Plaza, The k􀀀Fibonacci sequence and the Pascal 2-triangle, Chaos Solit. Fractals., 33 (2007), 38-49. $ \href{https://doi.org/10.1016/j.chaos.2006.10.022}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33846818865&origin=resultslist&sort=plf-f&src=s&sid=ee65ee18b37912212bc7c21b14ac96e3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fibonacci+sequence+and+the+Pascal+2-triangle%22%29&sl=62&sessionSearchId=ee65ee18b37912212bc7c21b14ac96e3&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000245570800005}{\mbox{[Web of Science]}} $
• [3] S. Falcon and A. Plaza, Binomial transforms of the k-Fibonacci sequence, Int. J. Nonlin. Sci. Num., 10 (11-12) (2009), 1527-1538. $ \href{https://doi.org/10.1515/IJNSNS.2009.10.11-12.1527}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-77950419067&origin=resultslist&sort=plf-f&src=s&sid=3f20e18152260993d6ece4f6a014bcbd&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Binomial+transforms+of+the+k-Fibonacci+sequence%22%29&sl=64&sessionSearchId=3f20e18152260993d6ece4f6a014bcbd&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276427900016}{\mbox{[Web of Science]}}$
• [4] P. Bhadouria, D. Jhala and B. Singh. Binomial transforms of the k-Lucas sequences and its properties, J. Math. Computer Sci., 8(1) (2014), 81-92. $ \href{http://dx.doi.org/10.22436/jmcs.08.01.07}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000219461500007}{\mbox{[Web of Science]}} $
• [5] M.Z. Spivey and L.L. Steil, The k􀀀Binomial transform and the Hankel transform, J. Integer Seq., 9(1) (2006), 1-19. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-27944507263&origin=resultslist&sort=plf-f&src=s&sid=3f20e18152260993d6ece4f6a014bcbd&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+k-Binomial+transform+and+the+Hankel+transform%22%29&sl=64&sessionSearchId=3f20e18152260993d6ece4f6a014bcbd&relpos=0}{\mbox{[Scopus]}} $
• [6] Y. Yazlık, N. Yılmaz and N. Tas¸kara, The generalized (s; t)􀀀matrix sequence’s binomial transforms, Gen. Math. Notes, 24(1) (2014), 127- 136. $ \href{http://www.kurims.kyoto-u.ac.jp/EMIS/journals/GMN/yahoo_site_admin/assets/docs/12_GMN-6152-V24N1.293225238.pdf}{\mbox{[Web]}} $
• [7] N. Yılmaz and N. Tas¸kara, Binomial transforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., 2013(2013), 1-7. $ \href{https://doi.org/10.1155/2013/497418}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84888315749&origin=resultslist&sort=plf-f&src=s&sid=3f20e18152260993d6ece4f6a014bcbd&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Binomial+transforms+of+the+Padovan+and+Perrin+matrix+sequences%22%29&sl=64&sessionSearchId=3f20e18152260993d6ece4f6a014bcbd&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000326536300001}{\mbox{[Web of Science]}} $
• [8] J.P. Allouche and J. Johnson, Narayana’s cows and delayed morphisms, Journees d’Informatique Musicale (JIM96), May(1996), France, 2-7. $ \href{https://hal.science/hal-02986050/document}{\mbox{[Web]}} $
• [9] T.V. Didkivska and M.V. St’opochkina, Properties of Fibonacci-Narayana numbers, World Math., 9(1) (2003), 29-36.
• [10] G. Bilgici, The generalized order-k Narayana’s cows numbers, Math. Slovaca, 66(4) (2016), 795–802. $ \href{https://doi.org/10.1515/ms-2015-0181}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84994440934&origin=resultslist&sort=plf-f&src=s&sid=3f20e18152260993d6ece4f6a014bcbd&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+generalized+order-k+Narayana%27s+cows+numbers%22%29&sl=64&sessionSearchId=3f20e18152260993d6ece4f6a014bcbd&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000387226200004}{\mbox{[Web of Science]}} $
• [11] Y. Soykan, On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech., 7(3) (2020), 43-56. $ \href{http://www.ijaamm.com/uploads/2/1/4/8/21481830/v7n3p5_43-56.pdf}{\mbox{[Web]}} $
• [12] J.L. Ram´ırez and V.F.Sirvent, A note on the k-Narayana sequence, Ann. Math. Inform., 45 (2015), 91-105. $\href{https://www.webofscience.com/wos/woscc/full-record/WOS:000434915800009}{\mbox{[Web of Science]}} $
• [13] S. Uygun and A. Erdoğdu, Binomial transforms k-Jacobsthal sequences, J. Math. Comput. Sci., 7(6) (2017), 1100-1114. $ \href{https://doi.org/10.28919/jmcs/3474}{\mbox{[CrossRef]}} $
• [14] S. Uygun, The binomial transforms of the generalized (s,t)-Jacobsthal matrix sequence, Int. J. Adv. Appl. Math. Mech., 6(3) (2019), 14-20. $\href{http://www.ijaamm.com/uploads/2/1/4/8/21481830/v6n3p2_14-20.pdf}{\mbox{[Web]}} $
• [15] N. Tas¸kara, K. Uslu and H.H. Gulec¸, On the properties of Lucas numbers with binomial coefficients, Appl. Math. Lett., 23(1) (2010), 68-72 . $ \href{https://doi.org/10.1016/j.aml.2009.08.007}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-70350622157&origin=resultslist&sort=plf-f&src=s&sid=7ccb958c825c06c322662e3ddf32d867&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+properties+of+Lucas+numbers+with+binomial+coefficients%22%29&sl=78&sessionSearchId=7ccb958c825c06c322662e3ddf32d867&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000272642100015}{\mbox{[Web of Science]}} $
• [16] H. Prodinger, Some information about the binomial transform, Fibonacci Quart., 32(5) (1994), 412-415. $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1994PM48100008}{\mbox{[Web of Science]}} $
• [17] K.W. Chen, Identities from the binomial transform, J. Number Theory, 124(1) (2007), 142-150. $ \href{https://doi.org/10.1016/j.jnt.2006.07.015}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33847091204&origin=resultslist&sort=plf-f&src=s&sid=d013993b06c44ad58f9819662d9071f4&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Identities+from+the+binomial+transform%22%29&sl=62&sessionSearchId=d013993b06c44ad58f9819662d9071f4&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000246235800009}{\mbox{[Web of Science]}} $