Abstract

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.

1. Introduction

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

2. The k-Lucas Numbers and Properties

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:

3. On the Symmetric Functions of Some Numbers and Polynomails

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 function 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

4. Conclusion

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.

Acknowledgments

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

References