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Research Article

Open Access Peer-reviewed

Khadidja Boubellouta^{ }, Ali Boussayoud, Mohamed Kerada

Received January 19, 2018; Revised March 22, 2018; Accepted June 21, 2018

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.

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 t**wo** 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*

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) |

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 genera**ting 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}

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

[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

This 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/

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

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.

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.

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 | |||