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
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With or 1,
![]() |
With
First, we set and
(by convention).
For or
, we set
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Definition 1. 1 Let and
be any two alphabets, then we give
by the following form:
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with the condition for
Definition 2. 2 Let be any function on
then we consider the divided difference operator as the following form
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Definition 3. 7 The symmetrizing operator is defined by
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Remark 1. Let an alphabet, we have
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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:
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• If the Pell-Lucas numbers appears:
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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
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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
![]() |
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 | ||
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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 | ||