Abstract

In this paper, we derive new generating functions for the products of k-Fibonacci numbers, k-Pell numbers, k-Jacobsthal numbers and the Chebychev polynomials of the second kind by making use of useful properties of the symmetric functions.

1. Introduction and Preliminaries

Fibonacci and Lucas numbers have been studied by many researchers for a long time to get intrinsic theory and applications of these numbers in many research areas as Physics, Engineering, Architecture, Nature and Art. For example, the ratio of two consecutive numbers converges to the Golden ratio which was thoroughly interested in ¹. We should recall that , for , k-Fibonacci and k-Jacobsthal sequences have been defined by the recursive equations ^{2, 3};

and

For the special case , it is clear that these two sequences are simplified to the well-known Fibonacci and Jacobsthal sequences, respectively. More recently, many papers are dedicated to Fibonacci sequence, such as the works of Caldwell et al. in ⁴, Marques in ⁵, Shattuck in ⁶ and Falcon et al. in ⁷.

The main purpose of this paper is to present some results involving the k-Fibonacci and k-Jacobsthal numbers using define a new useful operator denoted by By making use of this operator, we can derive new results based on our previous ones ^{8, 9, 10, 11, 12}. In order to determine generating functions of the product of k-Fibonacci and k-Jacobsthal numbers and Chebychev polynomials of second kind, we combine between our indicated past techniques and these presented polishing approaches.

Here, we recall some basic definitions and theorems that are needed in the sequel.

Definition 1. ² Let A and B be any two alphabets, then we give by the following form

(1)

with the condition for .

Definition 2. ¹³ Taking in (1) gives

(2)

Definition 3. ¹⁴ Given a function on the divided difference operator is defined as follows

Definition 4. ¹⁵ The symmetrizing operator is defined by

for all .

Remark 1. If , we have

2. Main Results

In this section, we combine all results obtained here in a unified way such that they can be considered as special cases of the following Theorems.

Theorem 1. Given two alphabets and then

(3)

Proof. Let and be two sequences such that .

On one hand, since and , we have

which is the left hand side of (3). On the other hand, since

we have that

So, this completes the proof.

Theorem 2. ¹⁶ Given two alphabets and we have:

(4)

Theorem 3. Given two alphabets and we have

(5)

3. On The Symmetric and Generating Functions

In this section, the new generating functions of the products of k-Fibonacci numbers, k-Pell numbers, k-Jacobsthal numbers and the Chebychev polynomials of the second kind are given by using the previous theorems.

Case 1: Replacing by and by in (5) yields

(6)

This case consists of two related parts.

Firstly, the substitutions of

In (6), we deduce the following theorem

Theorem 4. We have the following a new generating function of the product of k-Fibonacci numbers and k-Pell numbers is given by

(7)

Corollary 1. If in the relationship (7) we get

which represents a generating function of the product of Fibonacci numbers and Pell numbers ¹⁷.

Secondly, the substitution of

in (6) yields

We deduce the following theorem.

Theorem 5. For the new generating function of the product of k-Pell numbers and k-Jacobsthal numbers is given by

(8)

Corollary 2. If in the relationship (8) we get

which represents a generating function of the product of Pell numbers and Jacobsthal numbers ¹⁷.

Case 2. Replacing by and by in (4) yields

(9)

The substitution of

in (9), we deduce the following theorem.

Theorem 6. For the new generating function of the produce of k-Fibonacci numbers and k-Jacobsthal numbers is given by

(10)

Corollary 3. In the special case identity (10) gives

which represents a generating function of the product of Fibonacci numbers and Jacobsthal numbers ¹⁷.

Case 3: Remplacing by , by and by in (3) yields

(11)

This case consists of three related parts.

Firstly, the substitutions of in (11), we deduce the following theorem.

Theorem 7. ² We have a generating function of the product of k-Fibonacci numbers and Chebychev polynomial of the second kind

(12)

Secondly, the substitution of in (11), we deduce the following theorem.

Theorem 8. For, the new generating function of the product of k-Pell numbers and Chebychev polynomial of the second kind is given by

(13)

• if in the relationship (13) we get ²

which represents a generating function of the product of Pell numbers and Chebychev polynomial of the second kind.

Finally, the substitution of in (11) gives

We deduce the following theorem.

Theorem 9. We obtain a new generating function of the product of k-Jacobsthal numbers and Chebychev polynomial of the second kind as

(14)

Corollary 4. If in the relationship (14) we get

which represents a new generating function of the product of Jacobsthal numbers and Chebychev polynomial of the second kind.

Case 4. Replacing by and by in (3) and (5) yields

(15)

(16)

This case consists of three related parts.

Firstly, the substitution of

In (15) and (16), we obtain

(17)

(18)

Multiplying the equation (17) by 2 and added to (18) by -1, we obtain

(19)

which represents a new generating function of the product of k-Fibonacci numbers and k-Lucas numbers.

• For in (19) we obtain

which represents a generating function of even indices of Fibonacci numbers ¹⁷.

Secondly, the substitution of

in (15) and (16), we get

(20)

(21)

Multiplying the equation (20) by 2 and added to (21) by -2, we have

(22)

which represents a new generating function of the product of k-Pell numbers and k-Pell-Lucas numbers.

Corollary 5. If in the relationship (22) we get

which represents a generating function of even indice of Pell numbers ¹⁷.

Thirdly, the substitution of

in (15) and (16), we get

(23)

(24)

Multiplying the equation (23) by 2 added to (24) by (-1), we obtain

(25)

which represents a new generating function of the product of k-Jacobsthal numbers and k-Jacobsthal-Lucas numbers.

Corollary 6. If in the relationship (25) we have

which represents a generating function of even indices for Jacobsthal numbers ¹⁷

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

References