VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 15 Vaishali Billore and Naresh Patel
1Department of Applied Mathematics, Institute of Engineering & Technology, Indore (M.P.) 452001, India
2Department of Mathematics, Government Holkar (Model, Autonomous) Science College, Indore (M.P.) 452001, India
Abstract - In this paper, two matrices are employed to locate the Extended Generalized Lucas polynomials elements. Our study is expanded to include Extended Generalized Lucas polynomials with n- steps. Following that, the paper generalizes the Lucas polynomials and their relationship. Additionally, we provide connections between the Extended generalized Lucas and Generalized Fibonacci polynomials.
Keywords: Fibonacci & Lucas polynomials, Generalized Fibonacci & Lucas polynomials, Extended Generalized Lucas polynomials.
1 INTRODUCTION
The use of Fibonacci numbers and sequences is really fascinating. Numerous mathematical subfields use the Fibonacci sequence. Calculus, applied mathematics, linear algebra, group theory, and other topics are some examples. Furthermore, there are numerous significant uses for these numbers in many different disciplines, including computer science, physics, biology, and statistics. Several generalised Fibonacci and Lucas sequences are also shown in [1,2,4,8,9,13,16] as well as applications of the Fibonacci sequence in group theory [3, 7].
Fibonacci polynomials are used in [6, 15, 18] in a few different ways. Hoggatt and Bicknell’s research on the Fibonacci polynomials and n-step Fibonacci polynomials and their presentation of some of the polynomials’ features [11] is one of the studies conducted in this field. E. Ozkan, et al. [14] established the connection between Lucas polynomials and Fibonacci polynomials after obtaining the terms of n-step Lucas polynomials using matrices and generalizing the idea. Although Singh et al. followed it for second order sequences, which can be further extended to higher order sequences and matrix construction, they offered a generalization of Fibonacci-Lucas sequences, produced a matrix of order three, and established several identities in [17].
In this article, we introduce an Extended generalized Lucas polynomials by aid of a new matrix and the generalized Q- Matrix investigated by G. Lee and M. Asci [10]. We extend this study to the n- step Extended generalized Lucas polynomials. Then the extended Lucas polynomials and their relationship are generalized in this paper. Further, we give relationships between generalized Fibonacci polynomials and generalized Lucas polynomials.
2 PRELIMINARIES
Definition 1. The Fibonacci polynomials are defined by recurrence relation
Definition 2. The Generalized Fibonacci polynomials called (p; q) Fibonacci polynomials are defined by the recurrence relation
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 16 with initial conditions
Table 1 The first few (p; q)- Fibonacci polynomials and the array of their coefficient
That the (p; q)- Fibonacci polynomials are generated by matrix Qp,q
It can be verified by mathematical induction. Also it is clear that if p(x) = 1 and q(x) = 1
becomes Fibonacci number.
Moreover, when we write pascal’s triangle in left-justified form, the sums of the elements along the rising diagonals bring coefficients of the (p; q)- Fibonacci polynomials. That is
where [x] is the greatest integer contained in x and is a binomial coefficient.
3 MAIN RESULTS
3.1 Extended Generalized Lucas Polynomials
Definition 3. The Extended Generalized Lucas polynomials are defined by the recurrence relation
Theorem 3.1. For
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 17
Proof. This can be verified by mathematical induction on n. For n = 1, the claim is true, Since
Suppose the claim is true for n = m, that is
we have to prove that the claim is true for n = m + 1. Hence
We can express the Extended generalized Lucas polynomials as follows:
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 18 where [x] is the greatest integer contained in x and is a binomial coefficient.
Lemma 3.1. We can write the following interesting determinant identity
Definition 4. The 3- step Extended generalized Lucas polynomials are defined by the recurrence relation
with initial conditions
When them becomes 3- step Lucas numbers
i.e. 3, 1, 3, 7, 11, 21, 39,……..
Theorem 3.2. For Then
Proof. This can be verified by mathematical induction.
For n = 1, it is apparent that
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 19 Now suppose the claim is true for n = m, i.e.
We have to prove that the claim is true for n = m + 1. Hence
The first few 3- step Extended generalized Lucas polynomials are given below
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 20
Definition 5. The 4- step Extended generalized Lucas polynomials are defined by the recurrence relation
The first few 4- step Extended generalized Lucas polynomials are given below:
Theorem 3.3. For we have
Proof. This can be verified by mathematical induction (as previous theorems).
Definition 6. The r- step Extended generalized Lucas polynomials are defined by the recurrence relation
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 21 Theorem 3.4. For We have
Proof. This can be verified by mathematical induction.
For n = 1, then the claim is true, Since
VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 22 Suppose the claim is true for
we have to prove that the claim is true for n = m + 1, we have
Hence,
Using above definition we have,
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3.2 Relationships between Generalized Fibonacci polynomials and Extended Generalized Lucas Polynomials
Theorem 3.5. The Generalized Fibonacci polynomials and Extended Generalized Lucas polynomials have the following relationship for any positive integer n:
Proof. From the Generalized Fibonacci polynomials definition, we derive the following equality,
From above, we have
Theorem 3.6. The Generalized Fibonacci polynomials and Extended Generalized Lucas polynomials have the following relationship for any positive integer n:
Proof. From the Generalized Fibonacci polynomials definition, we derive the following equality,
From above, we have
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Extended Generalized Lucas polynomials and gave relationship between Generalized Fibonacci and Extended Generalized Lucas polynomials.
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