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.

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

(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)

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

(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.

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.

References