Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2018, 6(3), 107-110
DOI: 10.12691/tjant-6-3-8
Open AccessResearch Article

The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences

Moussa Ahmia1, and Hacène Belbachir2

1Department of Mathematic, University of Mohamed Seddik Ben yahia, Jijel, Algeria

2Faculty of Mathematics, University of Sciences and Technology Houari Boumediene, Algiers, Algeria

Pub. Date: May 12, 2018

Cite this paper:
Moussa Ahmia and Hacène Belbachir. The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences. Turkish Journal of Analysis and Number Theory. 2018; 6(3):107-110. doi: 10.12691/tjant-6-3-8

Abstract

In this paper, we show the log-concavity properties for the hyperjacobsthal, hyperjacobsthal-Lucas and associated sequences. Further, we investigate the -log-concavity property.

Keywords:
Hyperjacobsthal numbers hyperjacobsthal-Lucas numbers log-concavity -log-concavity

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  F. Brenti, Log-concave and unimodal sequence in algebra, combinatorics and geometry: an update. Elec. Contemp. Math. 178 (1994, 1997), 71-84.
 
[2]  R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989), 500-534.
 
[3]  Y. Wang, Y.-N. Yeh, Log-concavity and LC-positivity, J. Combin. Theory Ser. A, 114 (2007), 195-210.
 
[4]  L. M. Butler, The q-log concavity of q-binomial coefficients, J. Combin. Theory Ser. A 54 (1990), 54-63.
 
[5]  W. Y. C. Chen, L. X. W. Wang and A. L. B. Yang, Schur positivity and the q-log-convexity of the Narayana polynomials, J. Algebr. Comb. 32 (2010), 303-338.
 
[6]  B.-X. Zhu, Log-convexity and strong q-log-convexity for some triangular arrays, Adv. in. Appl. Math. 50(4) (2013), 595-606.
 
[7]  N-N. Cao, F-Z. Zhao, Some Properties of Hyperfibonacci and Hy-perlucas Numbers, Journal of Integer Sequences, 13(8) (2010), Article 10.8.8.
 
[8]  A. Dil, I. Mezö, A symmetric algorithm for hyperharmonic and Fibonacci numbers,Appl. Math. Comput. 206 (2008), 942-951.
 
[9]  N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, http://oeis.org, (2014).
 
[10]  L.-N. Zheng, R. Liu, On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers, J. Integer Sequences, Vol. 17 (2014), Article 14.1.4.
 
[11]  M. Ahmia, H. Belbachir, A. Belkhir, The log-concavity and log-convexity properties associated to hyperpell numbers and hyperpell-lucas numbers, Annales Mathematicae et Informaticae. 43 (2014), 3-12.
 
[12]  A. F. Horadam. Jacobsthal Representation Numbers. Fibonacci Quarterly, 34 (1) (1996), 40-54.
 
[13]  A. F. Horadam. Jacobsthal and Pell Curves. The Fibonacci Quarterly 26.1 (1988), 79-83.
 
[14]  K. V. Menon. On the convolution of logarithmically concave sequences, Proc. Amer. Math. Soc, 23 (1969), 439-441.
 
[15]  D. W. Walkup, Pólya sequences, binomial convolution and the union of random sets, J. Appl. Probab, 13 (1976), 76-85.
 
[16]  M. Ahmia, H. Belbachir, Preserving log-concavity and general-ized triangles. T. Komatsu (ed.), Diophantine analysis and related fields 2010. NY: American Institute of Physics (AIP). AIP Conference Proceedings 1264 (2010), 81-89.
 
[17]  M. Ahmia, H. Belbachir, Preserving log-convexity for generalized Pascal triangles, Electron. J. Combin. 19(2) (2012), Paper 16, 6 pp.
 
[18]  F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. no. 413 (1989).
 
[19]  H. Davenport, G. Pólya, On the product of two power series, Canadian J. Math. 1 (1949), 1-5.
 
[20]  L. Liu, Y. Wang, On the log-convexity of combinatorial sequences, Advances in Applied Mathematics 39(4) (2007), 453-476.