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Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials

Khadidja Boubellouta , Ali Boussayoud, Mohamed Kerada
Turkish Journal of Analysis and Number Theory. 2018, 6(3), 98-102. DOI: 10.12691/tjant-6-3-6
Received January 19, 2018; Revised March 22, 2018; Accepted June 21, 2018

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

[1]  Koshy, T, “Fibonacci and Lucas Numbers With Applications,” Wiley-Interscience, 2001.
In article      View Article
 
[2]  Boussayoud, A. and Harrouche, N, “Complet Symmetric functions and k-Fibonacci Numbers,” Commun. Appl. Anal, 20.457-467. 2016.
In article      View Article
 
[3]  Yilmaz, F. and Bozkurt, D, “The Generalized Order-k Jacobsthal Numbers,” Int. J. Contemp.Math.Sciences. 34 Sciences, 1685-1694. 2009.
In article      View Article
 
[4]  Caldwell, C. k. and Komatsu, T, “Some Periodicities in the Continued fraction Expansion of Fibonacci and Lucas Dirichelet Series,” Fibonacci Quarterly, 48(1).47-55.2010.
In article      View Article
 
[5]  Marques, D, “The Order of Appearance of the Product of Consecutive Lucas Numbers,” The Fibonacci Quarterly, 51(1). 38-43. 2013.
In article      
 
[6]  Shattuck, M, “Combinatorial Proofs of Determinant Formulas for the Fibonacci and Lucas Polynomial,” The Fibonacci Quarterly, 51(1).63-71.2013.
In article      View Article
 
[7]  Falcόn, S and Plaza, Ả, “On the Fibonacci k-numbers,” Chaos, Solitons & Fractals, 32(5).1615-1624.2007.
In article      View Article
 
[8]  Boussayoud, A., kerada, M. and Harrouche, N,, “On the k-Lucas numbers and Lucas Polynomails,” Turkish Journal of Analysis and Number, 5(3). 121-125. 2017.
In article      View Article
 
[9]  Boussayoud, A., Abderrezzak, A, “On Some Identities and Generating Functions for Hadamard Product,” Electron. J. Math. Analysis Appl, 5 (2). 89-97.2017.
In article      View Article
 
[10]  Boussayoud, A., Bolyer, M. and Kerada, M, “On Some Identities and Symmetric Functions for Lucas and Pell numbers,” Electron. J. Math. Analysis Appl, 5 (1). 202-207. 2017.
In article      View Article
 
[11]  Boussayoud, A, “Symmetric functions for k-Pell Numbers at negative indices,” Tamap Journal of Mthematics and Statistics, 1ID20 .1-8. 2017.
In article      View Article
 
[12]  Boussayoud, A, “On some identities and generating functions for Pell-Lucas numbers,” Online .J. Anal. Comb, 12.1-10.2017.
In article      View Article
 
[13]  Boussayoud, A., Bouler, M. and Kerada, M, “A simple and accurate method for determination of some generalized sequence of numbers,” I nt. J. Pure Appl. Math, 108. 503-511. 2016.
In article      
 
[14]  Boussayoud, A., Abderrezzak, M.and Kerada, M, “Some applications of symmetric functions,” Integers, 15A#48.1-7.2015.
In article      
 
[15]  Boussayoud, A., Kerada, M., Sahali, R. and Rouibah, W,“Some Application on Generating Functions,”J. Concr. Appl. Math, 12.321-330.2014.
In article      
 
[16]  Boussayoud, A. and Kerada, M, “Symmetric and Generating Functions,” Int.Electron. J. Pure Appl. Math. 7. 195-203.2014.
In article      View Article
 
[17]  Mezo, I, “Several Generating Functions for Second-Order Recurrence Sequences,” J. Integer Seq, 12.1-16.2009.
In article      View Article
 
[18]  Bolat, C. and Kose, H, “On the Properties of k-Fibonacci Numbers,” Int . J. Contemp. Math. Sciences, 1097-1105. 2010.
In article      View Article
 
[19]  Hoggatt, V.E, “Fibonacci and Lucas Numbers,” A publication of the Fibonacci Association. University of Santa Clara, Santa Clara. Houghton Mifflin Company, 1969.
In article      
 
[20]  Vorobiov, N. N, “Némeors de Fibonacci,” Editora MIR, URSS, 1974.
In article      
 

Published with license by Science and Education Publishing, Copyright © 2018 Khadidja Boubellouta, Ali Boussayoud and Mohamed Kerada

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/

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Normal Style
Khadidja Boubellouta, Ali Boussayoud, Mohamed Kerada. Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials. Turkish Journal of Analysis and Number Theory. Vol. 6, No. 3, 2018, pp 98-102. http://pubs.sciepub.com/tjant/6/3/6
MLA Style
Boubellouta, Khadidja, Ali Boussayoud, and Mohamed Kerada. "Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials." Turkish Journal of Analysis and Number Theory 6.3 (2018): 98-102.
APA Style
Boubellouta, K. , Boussayoud, A. , & Kerada, M. (2018). Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials. Turkish Journal of Analysis and Number Theory, 6(3), 98-102.
Chicago Style
Boubellouta, Khadidja, Ali Boussayoud, and Mohamed Kerada. "Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials." Turkish Journal of Analysis and Number Theory 6, no. 3 (2018): 98-102.
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[1]  Koshy, T, “Fibonacci and Lucas Numbers With Applications,” Wiley-Interscience, 2001.
In article      View Article
 
[2]  Boussayoud, A. and Harrouche, N, “Complet Symmetric functions and k-Fibonacci Numbers,” Commun. Appl. Anal, 20.457-467. 2016.
In article      View Article
 
[3]  Yilmaz, F. and Bozkurt, D, “The Generalized Order-k Jacobsthal Numbers,” Int. J. Contemp.Math.Sciences. 34 Sciences, 1685-1694. 2009.
In article      View Article
 
[4]  Caldwell, C. k. and Komatsu, T, “Some Periodicities in the Continued fraction Expansion of Fibonacci and Lucas Dirichelet Series,” Fibonacci Quarterly, 48(1).47-55.2010.
In article      View Article
 
[5]  Marques, D, “The Order of Appearance of the Product of Consecutive Lucas Numbers,” The Fibonacci Quarterly, 51(1). 38-43. 2013.
In article      
 
[6]  Shattuck, M, “Combinatorial Proofs of Determinant Formulas for the Fibonacci and Lucas Polynomial,” The Fibonacci Quarterly, 51(1).63-71.2013.
In article      View Article
 
[7]  Falcόn, S and Plaza, Ả, “On the Fibonacci k-numbers,” Chaos, Solitons & Fractals, 32(5).1615-1624.2007.
In article      View Article
 
[8]  Boussayoud, A., kerada, M. and Harrouche, N,, “On the k-Lucas numbers and Lucas Polynomails,” Turkish Journal of Analysis and Number, 5(3). 121-125. 2017.
In article      View Article
 
[9]  Boussayoud, A., Abderrezzak, A, “On Some Identities and Generating Functions for Hadamard Product,” Electron. J. Math. Analysis Appl, 5 (2). 89-97.2017.
In article      View Article
 
[10]  Boussayoud, A., Bolyer, M. and Kerada, M, “On Some Identities and Symmetric Functions for Lucas and Pell numbers,” Electron. J. Math. Analysis Appl, 5 (1). 202-207. 2017.
In article      View Article
 
[11]  Boussayoud, A, “Symmetric functions for k-Pell Numbers at negative indices,” Tamap Journal of Mthematics and Statistics, 1ID20 .1-8. 2017.
In article      View Article
 
[12]  Boussayoud, A, “On some identities and generating functions for Pell-Lucas numbers,” Online .J. Anal. Comb, 12.1-10.2017.
In article      View Article
 
[13]  Boussayoud, A., Bouler, M. and Kerada, M, “A simple and accurate method for determination of some generalized sequence of numbers,” I nt. J. Pure Appl. Math, 108. 503-511. 2016.
In article      
 
[14]  Boussayoud, A., Abderrezzak, M.and Kerada, M, “Some applications of symmetric functions,” Integers, 15A#48.1-7.2015.
In article      
 
[15]  Boussayoud, A., Kerada, M., Sahali, R. and Rouibah, W,“Some Application on Generating Functions,”J. Concr. Appl. Math, 12.321-330.2014.
In article      
 
[16]  Boussayoud, A. and Kerada, M, “Symmetric and Generating Functions,” Int.Electron. J. Pure Appl. Math. 7. 195-203.2014.
In article      View Article
 
[17]  Mezo, I, “Several Generating Functions for Second-Order Recurrence Sequences,” J. Integer Seq, 12.1-16.2009.
In article      View Article
 
[18]  Bolat, C. and Kose, H, “On the Properties of k-Fibonacci Numbers,” Int . J. Contemp. Math. Sciences, 1097-1105. 2010.
In article      View Article
 
[19]  Hoggatt, V.E, “Fibonacci and Lucas Numbers,” A publication of the Fibonacci Association. University of Santa Clara, Santa Clara. Houghton Mifflin Company, 1969.
In article      
 
[20]  Vorobiov, N. N, “Némeors de Fibonacci,” Editora MIR, URSS, 1974.
In article