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

Moussa Ahmia*, Hacène Belbachir

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

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

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

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

1. Introduction

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.

2. Definitions

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

[picture] | (2.0)

See for instance [12].

The sequences and satisfy the following recurrences

[picture] | (2.1)

For more details, see for instance [13].

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

[picture] | (2.2)

[picture] | (2.3)

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

[picture] | (2.4)

[picture] | (2.5)

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

[picture] | (2.6)

and

[picture] | (2.7)

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

[picture] | (2.8)

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

[picture] | (2.9)

and

[picture] | (2.10)

3. The Log-concavity Property

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

[picture] | (3.1)

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

[picture] | (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

[picture] | (3.3)

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

[picture] | (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

[picture] | (3.5)

[picture] | (3.6)

[picture] | (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.

Proof. When ,

When , through computation, we get

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

4. Concluding remarks

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.

Acknowledgements

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

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