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Research Article

Open Access Peer-reviewed

Moussa Ahmia^{ }, Hacène Belbachir

Received January 20, 2018; Revised March 26, 2018; Accepted June 25, 2018

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

Let be a sequence of nonnegative numbers. If for all (respectively ), the sequence is called log-concave (respectively log-convex), which is equivalent to (respectively for .

The log-concave and log-convex sequences arise often in combinatorics, algebra, geometry, analysis, probability and statistics and have been extensively investigated. We refer the reader to ^{ 1, 2, 3} for log-concavity.

Let be a sequences of polynomials in . If for each , has nonnegative coefficients as a polynomials in ; we say that is -log-concave. The -log-concavity of polynomials have been extensively studied; see for instance ^{ 4, 5, 6}.

In ^{ 7, 8}, some properties of hyperfibonacci numbers and hyperlucas numbers are given. For Fibonacci numbers and Lucas numbers , it is well known that and are log-concave (see ^{ 9}).

L. Zheng and R. Liu ^{ 10} gived some properties of the hyperfibonacci numbers and hyperhucas numbers, and investigated the log-concavity and log-convexity of these numbers. Finaly, they also studied the log-concavity (log-convexity) of generalized hyperfibonacci numbers and hyperlucas numbers. In ^{ 11}, we established these properties for hyperpell numbers and hyperpell-Lucas numbers.

In section two, we give the definitions and some properties of hyperjacobsthal and hyperjacobsthal-Lucas sequences. In section three, we establish the generating functions of these sequences. In section 4, we discuss their log- concavity. In addition, we investigate the -log-concavity of some polynomials related to hyperjacobsthal and hyperjacobsthal-Lucas numbers.

Dil and Mezö ^{ 8} introduced the hyperfibonacci numbers and hyperlucas numbers to be

where is a positive integer, and and are Fibonacci and Lucas numbers, respectively.

**Definition 2.1.** Let be positive integer. The hyperjacobsthal numbers and hyperjacobsthal-Lucas numbers are defined as follows

where and and are Jacobsthal and Jacobsthal-Lucas sequences, respectively.

The initial values of and are as follows

Now we recall some formulas for Jacobsthal and Jacobsthal-Lucas numbers. It is well know that the Binet forms of and are

(2.0) |

See for instance ^{ 12}.

The sequences and satisfy the following recurrences

(2.1) |

For more details, see for instance ^{ 13}.

It follows from (2.1) that the following formulas hold:

(2.2) |

(2.3) |

It is easy to see, for example by induction, that

(2.4) |

(2.5) |

The generating function of Jacobsthall numbers and Jacobsthal-Lucas numbers, denoted and , are respectively

(2.6) |

and

(2.7) |

So, we establish the generating function of hyperjacobsthal and hyperjacobsthal-Lucas numbers using respectively

(2.8) |

The generating functions of hyperjacobsthal numbers and hyperjacobsthal-Lucas numbers are

(2.9) |

and

(2.10) |

We start the section by some useful lemmas.

**Lemma 3.1.**** **^{ 15} If the sequences and are log-concave, then so is their ordinary convolution

**Lemma 3.2.**** **^{ 15} If the sequences is log-concave, then so is the binomial convolution

The following result deals with the log-concavity of hyperjacobsthal and hyperjacobsthal-Lucas sequences.

**Theorem 3.****3****.**** **The sequences and are log-concave for and respectively.

**Proof.**** **To prove the results, we use the following relations

(3.1) |

When . When it follows from (2.2), (2.4), (2.5) and (3.1)

(3.2) |

There exist two cases. If n is even, then

else

Then is log-concave. By induction hypothesis and Lemma 3.1 the sequence is log-concave.

One can verify that

(3.3) |

It follows from (2.3), (2.5) and (3.3) that

(3.4) |

For , there exist two cases. If n is even, we get

else

Hence is log-concave. By induction hypothesis and Lemma 3.1 the sequence is log-concave. This completes the proof of Theorem 3.3.

Then we have the following corollary.

**Corollary 3.4.** The sequences and are log-concave for and respectively.

**Proof.**** **By Lemma 3.2.

Now we establish the log-concavity of order two of the sequences and for some special sub-sequences.

Theorem 3.5. Let

Then, the sub-sequences and are log-concave.

**Proof.**** **From (2.4), we get** **

(3.5) |

(3.6) |

(3.7) |

It follows form (3.2) and (3.5) that

Then is log-concave.

It follows from (3.2) and (3.6)

Then is log-concave.

Similarly, by applying (3.3) and (3.7), we have

Then is log-concave.

By same technic, we obtain

Then is log-concave. This completes the proof.

Then we have the following corollaries.

**Corollary 3.6.** The sequences and are log-concave.

**Proof.** By Lemma 3.2.

**Corollary 3.****7****.** The sequences and are log-concave.

**Proof.** By Lemma 3.2.

Now, we establish the -log-concavity property as follows.

**Theorem 3.****8****.**** **Define, for , the polynomials

The polynomials and are -log-concave for () and () respectively.

**P****roof. **When ,

When , through computation, we get

As () and () are log-concave, then the polynomials and are -log-concave for () and () respectively.

We have discussed the log-concavity of hyperjacobsthal numbers and hyperjacobsthal-Lucas numbers. In addition, we estabilished the -log-concavity of some polynomials related to the both numbers.

We would like to thank the referee for useful suggestions and several comments witch involve the quality of the paper.

[1] | F. Brenti, Log-concave and unimodal sequence in algebra, combinatorics and geometry: an update. Elec. Contemp. Math. 178 (1994, 1997), 71-84. | ||

In article | |||

[2] | R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989), 500-534. | ||

In article | View Article | ||

[3] | Y. Wang, Y.-N. Yeh, Log-concavity and LC-positivity, J. Combin. Theory Ser. A, 114 (2007), 195-210. | ||

In article | View Article | ||

[4] | L. M. Butler, The q-log concavity of q-binomial coeﬃcients, J. Combin. Theory Ser. A 54 (1990), 54-63. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[6] | B.-X. Zhu, Log-convexity and strong q-log-convexity for some triangular arrays, Adv. in. Appl. Math. 50(4) (2013), 595-606. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[8] | A. Dil, I. Mezö, A symmetric algorithm for hyperharmonic and Fibonacci numbers,Appl. Math. Comput. 206 (2008), 942-951. | ||

In article | View Article | ||

[9] | N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, https://oeis.org, (2014). | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[12] | A. F. Horadam. Jacobsthal Representation Numbers. Fibonacci Quarterly, 34 (1) (1996), 40-54. | ||

In article | View Article | ||

[13] | A. F. Horadam. Jacobsthal and Pell Curves. The Fibonacci Quarterly 26.1 (1988), 79-83. | ||

In article | View Article | ||

[14] | K. V. Menon. On the convolution of logarithmically concave sequences, Proc. Amer. Math. Soc, 23 (1969), 439-441. | ||

In article | View Article | ||

[15] | D. W. Walkup, Pólya sequences, binomial convolution and the union of random sets, J. Appl. Probab, 13 (1976), 76-85. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[17] | M. Ahmia, H. Belbachir, Preserving log-convexity for generalized Pascal triangles, Electron. J. Combin. 19(2) (2012), Paper 16, 6 pp. | ||

In article | View Article | ||

[18] | F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. no. 413 (1989). | ||

In article | |||

[19] | H. Davenport, G. Pólya, On the product of two power series, Canadian J. Math. 1 (1949), 1-5. | ||

In article | View Article | ||

[20] | L. Liu, Y. Wang, On the log-convexity of combinatorial sequences, Advances in Applied Mathematics 39(4) (2007), 453-476. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2018 Moussa Ahmia and Hacène Belbachir

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

Moussa Ahmia, Hacène Belbachir. The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 3, 2018, pp 107-110. https://pubs.sciepub.com/tjant/6/3/8

Ahmia, Moussa, and Hacène Belbachir. "The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences." *Turkish Journal of Analysis and Number Theory* 6.3 (2018): 107-110.

Ahmia, M. , & Belbachir, H. (2018). The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences. *Turkish Journal of Analysis and Number Theory*, *6*(3), 107-110.

Ahmia, Moussa, and Hacène Belbachir. "The Log-concavity Property Associated to Hyperjacobsthal and Hyperjacobsthal-Lucas Sequences." *Turkish Journal of Analysis and Number Theory* 6, no. 3 (2018): 107-110.

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[1] | F. Brenti, Log-concave and unimodal sequence in algebra, combinatorics and geometry: an update. Elec. Contemp. Math. 178 (1994, 1997), 71-84. | ||

In article | |||

[2] | R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989), 500-534. | ||

In article | View Article | ||

[3] | Y. Wang, Y.-N. Yeh, Log-concavity and LC-positivity, J. Combin. Theory Ser. A, 114 (2007), 195-210. | ||

In article | View Article | ||

[4] | L. M. Butler, The q-log concavity of q-binomial coeﬃcients, J. Combin. Theory Ser. A 54 (1990), 54-63. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[6] | B.-X. Zhu, Log-convexity and strong q-log-convexity for some triangular arrays, Adv. in. Appl. Math. 50(4) (2013), 595-606. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[8] | A. Dil, I. Mezö, A symmetric algorithm for hyperharmonic and Fibonacci numbers,Appl. Math. Comput. 206 (2008), 942-951. | ||

In article | View Article | ||

[9] | N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, https://oeis.org, (2014). | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[12] | A. F. Horadam. Jacobsthal Representation Numbers. Fibonacci Quarterly, 34 (1) (1996), 40-54. | ||

In article | View Article | ||

[13] | A. F. Horadam. Jacobsthal and Pell Curves. The Fibonacci Quarterly 26.1 (1988), 79-83. | ||

In article | View Article | ||

[14] | K. V. Menon. On the convolution of logarithmically concave sequences, Proc. Amer. Math. Soc, 23 (1969), 439-441. | ||

In article | View Article | ||

[15] | D. W. Walkup, Pólya sequences, binomial convolution and the union of random sets, J. Appl. Probab, 13 (1976), 76-85. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[17] | M. Ahmia, H. Belbachir, Preserving log-convexity for generalized Pascal triangles, Electron. J. Combin. 19(2) (2012), Paper 16, 6 pp. | ||

In article | View Article | ||

[18] | F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. no. 413 (1989). | ||

In article | |||

[19] | H. Davenport, G. Pólya, On the product of two power series, Canadian J. Math. 1 (1949), 1-5. | ||

In article | View Article | ||

[20] | L. Liu, Y. Wang, On the log-convexity of combinatorial sequences, Advances in Applied Mathematics 39(4) (2007), 453-476. | ||

In article | View Article | ||