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Research Article

Open Access Peer-reviewed

Ali Boussayoud^{ }, Mohamed Kerada, Nesrine Harrouche

Received December 23, 2016; Revised May 11, 2017; Accepted June 07, 2017

In this paper, we introduce an operator in order to derive some new symmetric properties of -Lucas numbers and Lucas polynomials. By making use of the operator defined in this paper, we give some new generating functions for -Lucas numbers and Lucas polynomials.

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 ^{ 13}. We should recall that, for , *k*-Fibonacci and *k*-Lucas sequences have been defined by the recursive equations ^{ 9, 10};

with initial conditions and respectively. For the special case *k=*1, it is clear that these two sequences are simplified to the well-known Fibonacci and Lucas sequences, respectively. In this contribution, we shall define a new useful operator denoted by for which we can formulate, extend and prove new results based on our previous ones ^{ 4, 5, 6}. In order to determine generating functions for *k*-Fibonacci numbers, *k* -Lucas numbers and Lucas polynomials, we combine between our indicated past techniques and these presented polishing approaches.

Let *k* and be two positive integers and are set of given variables, recall ^{ 8} that the *k* -th elementary symmetric function and the *k* -th complete homogeneous symmetric function are defined respectively by

With or 1,

With

First, we set and (by convention).

For or , we set

**Definition 1.** ^{ 1} *Let ** and ** be any two alphabets*,* then we give ** by the following form*:

with the condition for

**Definition 2. **^{ 2} *Let ** be any function on ** then we consider the divided difference operator as the following form*

**Definition 3. **^{ 7} *The symmetrizing operator ** is defined by*

**Remark** **1**.* **Let ** an alphabet*,* we have *

The *k* -Lucas numbers have been defined in ^{ 11} for any number *k* as follows.

**Definition 4. **^{ 11} *For any positive real number, the** -Lucas numbers, say ** **is defined recurrently by*

(2.1) |

*with initial conditions*

Note that if is a real variable then and they correspond to the Lucas polynomials defined by

Particular cases of the -Lucas numbers are

• If the classical Lucas numbers is obtained:

• If the Pell-Lucas numbers appears:

The well-known Binet's formula in the Lucas numbers theory allows us to express the -Lucas number in function of the roots and of the characteristic equation, associated to the recurrence relation (2.1):

(2.2) |

**Proposition 1.*** **(Binet's formula) The nth **k** -Lucas number is given by*

*where ** are the roots of the characteristic equation (2.2) and *

* Proof* . The roots of the characteristic equation (2.2) are and

Note that, since , the and and

If denotes the positive root of the characteristic equation, the general term may be written in the form ^{ 10}

and the limit of the quotient of two terms is

In addition, the general term of the -Lucas numbers may be obtained by the formula ^{ 10}:

**Theorem 1**. ^{ 4} *Let ** and ** be two alphabets, respectively,** **and ** , then we have *

(3.1) |

In the case based on Theorem 1, we deduce the following Lemmas.

**Lemma 1.** *Given an alphabet **, we have*

(3.2) |

* Proof.* Let and be two sequences such that the left-hand side of the formula (3.2) can be written as:

white the right -hand side can be expressed as

This completes the proof.

**Lemma 2**. *Given an alphabet** ** , we have*

(3.3) |

* Proof.* Let and be two sequences such that the left-hand side of the formula (3.3) can be written as:

white the right-hand side can be expressed as:

This completes the proof.

Taking and in (3.2) and (3.3), we obtain the generating functions given by Boussayoud et al ^{ 5} which arises

1) The generating function of the Fibonacci numbers

2) The generating function of the Lucas numbers

**Proposition 2.** ^{ 10, 12}* The relations*

1)

2)

*hold for all*

Choosing and such that and substituting in (3.2) and (3.3) we end up with

(3.4) |

which represents a generatings functions for *k* -Fibonacci numbers (with and ).

(3.5) |

which represents a new generatings functions.

• Multiplying the equation (3.4) by and subtract it from (3.5) by , we obtain

from which we have the following theorem.

**Theorem 3.** *For **, the generating function of the ** -Lucas numbers is given by *

(3.6) |

• Put in the relationship (3.6) we have

which represents a generating function for Pell-Lucas numbers ^{ 5}.

Replacing *t* by in (3.4) and (3.6), we have the following theorems.

**Theorem 4**. *We have the following a new generating fu**nction of the **k** -Fibonacci numbers at negative indices as*

* Proof*. The ordinary generating function associated is defined by

Using the initial conditions, we get

Consider that and . Then can be written by

which is equivalent to

Replacing by , we have

therefore

This completes the proof.

**Theorem**** ****5**. *We have the following a new generating function of the ** -Lucas numbers at negative indices as*

(3.7) |

• Put in the relationship (3.7) we have

which represents a generating function for Pell-Lucas numbers at negative indices ^{ 3}.

Choosing and such that and substituting in (3.2) and (3.3) we end up with

(3.8) |

which represents a generating function of the Fibonacci polynomials

(3.9) |

which represents a new generatings functions.

• Multiplying the equation (3.8) by and subtract it from (3.9) by , we obtain

Thus we get the following theorem.

**Theorem 6**. *We have the following a generating function of the Lucas polynomials as*

(3.10) |

* Proof*. The ordinary generating function associated is defined by

Using the initial conditions, we get

Consider that and. Then can be written by

which is equivalent to

This completes the proof.

Replacing by in (3.8) and (3.10), we have the following theorems.

**Theorem 7**. *We have the following a new generating function of the Fibonacci polynomials at negative coefficient as*

**Theorem 8**. *We have the following a new generating function of the Lucas polynomials at negative coefficient as *

In this paper, a new theorem has been proposed in order to determine the generating functions. The proposed theorem is based on the symmetric functions. The obtained results agree with the results obtained in some previous works.

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

[1] | A. Abderrezzak, Généralisation de la transformation d'Euler d'une série formelle, Adv. Math. 103, 1994, 180-195. | ||

In article | View Article | ||

[2] | A. Abderrezzak, M. Kerada, A. Boussayoud, Generalization of Some Hadamard Product, Commun. Appl. Anal. 20(3), 301-306, (2016). | ||

In article | View Article | ||

[3] | A. Boussayoud, M. Boulyer, M. Kerada, On Some Identities and Symmetric Functions for Lucas and Pell Numbers, Electron .J.Math. AnalysisAppl. 5, 2017, 202-207. | ||

In article | View Article | ||

[4] | A. Boussayoud, N. Harrouche, Complete Symmetric Functions and k - Fibonacci Numbers, Commun. Appl. Anal. 20(4), 2016, 457-467. | ||

In article | View Article | ||

[5] | A. Boussayoud, M. Kerada, M. Boulyer, A simple and accurate method for determination of some generalized sequence of numbers, Int. J. Pure Appl. Math.108, 2016, 503-511. | ||

In article | |||

[6] | A. Boussayoud, A. Abderrezzak, M. Kerada, Some applications of symmetric functions, Integers. 15, A#48, 2015, 1-7. | ||

In article | |||

[7] | A. Boussayoud, M. Kerada, A. Abderrezzak, A Generalization of Some Orthogonal Polynomials, Springer Proc. Math. Stat. 41, 2013, 229-235. | ||

In article | |||

[8] | M. Merca, A Generalization of the symmetry between complete and elementary symmetric functions, Indian J. Pure Appl. Math. 45, 2014, 75-89. | ||

In article | View Article | ||

[9] | S. Falcon, A. Plaza, On the Fibonacci k - numbers, Chaos, Sulutions & Fractals. 32, 2007, 1615-1624. | ||

In article | View Article | ||

[10] | S. Falcon, On the k -Lucas Numbers of Arithmetic Indexes, Appl. Math. 3, 2012, 1202-1206. | ||

In article | View Article | ||

[11] | S. Falcon, On the k-Lucas numbers, J.Math.Comput.Sci. 2, 2012, 425-434. | ||

In article | |||

[12] | S. Falcon, On the complex k-Fibonacci numbers, Cogent Math. 3, 2016, 1-9. | ||

In article | View Article | ||

[13] | T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001. | ||

In article | View Article | ||

[14] | Y. K.Gupta, V. H. Badshah, M. Singh, K. Sisodiya, Some Identities of Tribonacci Polynomials, Turkish Journal of Analysis and Number Theory. 4, 2016, 20-22. | ||

In article | View Article | ||

[15] | Y. K.Gupta, V. H. Badshah, M. Singh, K. Sisodiya, Diagonal Function of k-Lucas Polynomials, Turkish Journal of Analysis and Number Theory. 3, 2015, 49-52. | ||

In article | View Article | ||

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

Ali Boussayoud, Mohamed Kerada, Nesrine Harrouche. On the *k* -Lucas Numbers and Lucas Polynomials. *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 4, 2017, pp 121-125. https://pubs.sciepub.com/tjant/5/4/1

Boussayoud, Ali, Mohamed Kerada, and Nesrine Harrouche. "On the *k* -Lucas Numbers and Lucas Polynomials." *Turkish Journal of Analysis and Number Theory* 5.4 (2017): 121-125.

Boussayoud, A. , Kerada, M. , & Harrouche, N. (2017). On the *k* -Lucas Numbers and Lucas Polynomials. *Turkish Journal of Analysis and Number Theory*, *5*(4), 121-125.

Boussayoud, Ali, Mohamed Kerada, and Nesrine Harrouche. "On the *k* -Lucas Numbers and Lucas Polynomials." *Turkish Journal of Analysis and Number Theory* 5, no. 4 (2017): 121-125.

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[1] | A. Abderrezzak, Généralisation de la transformation d'Euler d'une série formelle, Adv. Math. 103, 1994, 180-195. | ||

In article | View Article | ||

[2] | A. Abderrezzak, M. Kerada, A. Boussayoud, Generalization of Some Hadamard Product, Commun. Appl. Anal. 20(3), 301-306, (2016). | ||

In article | View Article | ||

[3] | A. Boussayoud, M. Boulyer, M. Kerada, On Some Identities and Symmetric Functions for Lucas and Pell Numbers, Electron .J.Math. AnalysisAppl. 5, 2017, 202-207. | ||

In article | View Article | ||

[4] | A. Boussayoud, N. Harrouche, Complete Symmetric Functions and k - Fibonacci Numbers, Commun. Appl. Anal. 20(4), 2016, 457-467. | ||

In article | View Article | ||

[5] | A. Boussayoud, M. Kerada, M. Boulyer, A simple and accurate method for determination of some generalized sequence of numbers, Int. J. Pure Appl. Math.108, 2016, 503-511. | ||

In article | |||

[6] | A. Boussayoud, A. Abderrezzak, M. Kerada, Some applications of symmetric functions, Integers. 15, A#48, 2015, 1-7. | ||

In article | |||

[7] | A. Boussayoud, M. Kerada, A. Abderrezzak, A Generalization of Some Orthogonal Polynomials, Springer Proc. Math. Stat. 41, 2013, 229-235. | ||

In article | |||

[8] | M. Merca, A Generalization of the symmetry between complete and elementary symmetric functions, Indian J. Pure Appl. Math. 45, 2014, 75-89. | ||

In article | View Article | ||

[9] | S. Falcon, A. Plaza, On the Fibonacci k - numbers, Chaos, Sulutions & Fractals. 32, 2007, 1615-1624. | ||

In article | View Article | ||

[10] | S. Falcon, On the k -Lucas Numbers of Arithmetic Indexes, Appl. Math. 3, 2012, 1202-1206. | ||

In article | View Article | ||

[11] | S. Falcon, On the k-Lucas numbers, J.Math.Comput.Sci. 2, 2012, 425-434. | ||

In article | |||

[12] | S. Falcon, On the complex k-Fibonacci numbers, Cogent Math. 3, 2016, 1-9. | ||

In article | View Article | ||

[13] | T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001. | ||

In article | View Article | ||

[14] | Y. K.Gupta, V. H. Badshah, M. Singh, K. Sisodiya, Some Identities of Tribonacci Polynomials, Turkish Journal of Analysis and Number Theory. 4, 2016, 20-22. | ||

In article | View Article | ||

[15] | Y. K.Gupta, V. H. Badshah, M. Singh, K. Sisodiya, Diagonal Function of k-Lucas Polynomials, Turkish Journal of Analysis and Number Theory. 3, 2015, 49-52. | ||

In article | View Article | ||