d-Tribonacci Polynomials and Their Matrix Representations
BAHAR KULOĞLU
1
, ENGIN ÖZKAN
2
1
Department of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali
Yıldırım University, Yalnızbağ Campus, 24100, Erzincan,
TURKEY
2
Department of Mathematics, Faculty of Arts and Sciences, Erzincan Binali Yıldırım University,
Yalnızbağ Campus, 24100, Erzincan,
TURKEY
Abstract: - In this study, we define d-Tribonacci polynomials. Some combinatorial properties of the d-
Tribonacci polynomials with matrix representations are obtained with the help of Riordan arrays. In addition, d-
Tribonacci number sequence, a new generalization of this number sequence, is obtained by considering the
Pascal matrix. With the help of the Pascal matrix, two kinds of factors of d-Tribonacci polynomials are found.
Also, infinite d-Tribonacci polynomials matrix and the inverses of these polynomials are found.
Key-Words: - d-Tribonacci polynomials, Generating function, Pascal matrix, Riordan matrix.
Received: July 18, 2022. Revised: January 16, 2023. Accepted: February 13, 2023. Published: March 23, 2023.
1 Introduction
The Tribonacci number sequence is inspired by the
Fibonacci number sequence and is a number
sequence with 3-term recurrence. It is used in many
branches, as in the Fibonacci number sequence.
Many generalizations of this number sequence such
as Padovan, Narayana, Perrin have been put forward
and studied [ 1-8, 10-12].
The term Tribonacci was first used by Feinberg in
1963 [14]. Later, many basic features were studied
[15-19].
We know that the Tribonacci numbers
are
defined by
,3
with
0,
0 and
1 [9].
In this study, a new Tribonacci number sequence is
obtained with the help of Riordan sequence and
Pascal matrix by bringing a new perspective to the
existing definitions of traditional number sequences.
Additionally, based on Pascal's matrix, we factor
two types of d-Tribonacci polynomials.
Also, infinite d-Tribonacci polynomial matrices and
the inverses of these polynomials are found.
It is thought that if these values are placed in the
Riordan array appropriately by working on the
initial values, it will allow similar studies to be
made on many number sequences where a Riordan
array is given as an infinite lower triangular matrix
if its th column generating function
is for . Note that the first column
is indexed by 0 and we accept [13].
Throughout this paper, let
and
be
polynomials with real coefficient for 1,…,
1.
Definition 1.1 d-Fibonacci polynomials are given
as:
⋯
(1)
with for and [12].
Similarly, d-Lucas polynomials are defined by
⋯
(2)
with
0for0and
2and
[12].
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The Riordan matrices is given as a set of matrices
where
are complex
numbers [13].
The Riordan group is defined as a set of infinite
lower-triangular integer matrices where each matrix
is defined by pair of formal power series
and with
and [13].
In this study, we describe new generalizations of
Tribonacci polynomials. Some combinatorial
properties of matrix representations of d-Tribonacci
polynomials are obtained with the help of Riordan
arrays. In addition, d-Tribonacci number sequence is
obtained by considering the Pascal matrix. Based on
the Pascal matrix, d-Tribonacci polynomials have
two types of factors.. Also, infinite d-Tribonacci
polynomial matrices and the inverses of these
polynomials are given.
2 Generalization of Tribonacci
Polynomials
Definition 2.1. Tribonacci polynomials are
given by
⋯
(3)
with
0,
1,
1 and
0 for 0.
A few terms of these polynomials:
0,
1,
1,
,
2
From equation (3), its characteristic equation are
obtained as
⋯
0.
Its roots:
,
,…,
.
Theorem 2.3. Generating function of
d‐
Tribonacci
polynomials
is
,
1
⋯
.
Proof. We have
,
⋯ (4)
Multiply Eq. (4) by
,
,…,
, respectively.
The following equations are obtained.
,
⋯
⋮
,
⋯
If the necessary calculations are made, we get
,1
⋯
0⋯.
,
⋯
.
Its Binet formula has the following form
We get the following equation for each value of .
⋮
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Multiplying both sides of above equations by
,
,…,
, respectively, we have:
⋮
The sum of the left-hand side of the equations:
1
⋯
.
The sum of the right-hand side of the equations:
1
⋯
1
1
so, we get
1
⋯
1
.
Theorem 2.4. We have the following equation for 0.
⋯
,
,…,
…
⋯
.
Proof. Generating function for Tribonacci polynomials
1
⋯
⋯
2
,
,…,
…
⋯
⋯
⋯
,
,…,
…
⋯
as desired.
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Theorem 2.5. The sum of the Tribonacci
polynomials:
1
1⋯
.
Proof. We have
⋯⋯
Multiplying the last equation by ,…,,
respectively then we obtain
⋯⋯
⋮
⋯⋯
From here, we have
1⋯1.
So, we get
1
1⋯
.
From [12], the Fibonacci polynomials matrix
has the following form
⋯
100
0⋱
⋱
0010
(5)
where 1.
Matrix representation for is given in
following theorem.
Theorem 2.7. The representation for is as
follows:
⋯⋯
⋯⋯
⋮⋮⋮
⋯⋯6
Proof. Let’s apply the induction over to prove it.
For 1,
⋯⋯
⋯⋯
⋮⋮⋮
⋯⋯
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⋯
100
0⋱
⋱
0010
(7)
From the definition of
, the matrices in (5) and (7) are equal.
Suppose that the result satisfies for . So, we obtain
⋯
⋯
⋯
⋯
⋮⋮⋮
⋯
⋯
Let’s prove it for 1. So, we get
⋯
⋯
⋯
⋯
⋮⋮⋮
⋯
⋯
.
⋯
100
0⋱
⋱
0010
⋯
⋯
⋯
⋯
⋮⋮⋮
⋯
⋯
Corollary 2.8. For ,0, we have
⋯
⋯
Proof. We know
.
The first row and column of matrix
is the
result.
Lemma 2.9. For 1,
.
Proof. For 2 equality is true
1
Let the equality be true for . For 1,
we show that the equation is true.
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⋯
, ⋯
.
Theorem 2.10. For 2,0,
⋯
,,…, …
,,…,
⋯ ⋯
(8)
Proof. For 1, we have
⋯
.
Let us show the right-hand side of (8) by RH.
For 0, we have
⋯
,,…, …
,,…,
⋯ ⋯
⋯
,,…, …
,,…,
⋯ ⋯
⋯
,,…, …
,,…,
⋯
⋯ ⋯
,,…, …
,,…,
⋯
⋯
⋯
⋯
from characteristic equation, we obtain
∑
.
as desired.
Lemma 2.11. For 1,
.
Proof. From (2) we get
⋯
⋯
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3 The Infinite Tribonacci Polynomials
Matrix
The Tribonacci polynomials matrix is showed
by
,
,…,
,,
and defined as follows
1
⋮
0.
1.
.
..
,
,
where
2
and
This Tribonacci polynomial matrix can also be
written as,
00…
0…
…
⋮⋮⋮⋮⋮…
Note that is a Riordan matrix.
Theorem 3.1. The first column of matrix is
1,
,
,…
.
From the Riordan group theory, we get the
generator function of the first column as follows:
,
,…,
,,
1
1
⋯
.
Proof. Generating functions of the first column of
matrix
is
1
⋯.
If we do operations like the proof of Theorem 2.4,
then
⋯
.
The desired expression is obtained.
From the Riordan matrix,
.
,
1
1
⋯
,.
If the Tribonacci polynomials matrix is
finite, then the matrix is
0000…
1000…
⋮⋮⋮⋮⋮…
⋮⋮
and
1
1.
We give two factorizations of Pascal Matrix with
the Tribonacci polynomials matrix. Now, we
give a matrix
,
,
,
1
1
2
1⋯
2
1
So, we get
100…
1 1 0 …
1 2 1 …
1 32 3 …
⋮⋮⋮…
Thus we can introduce the first factorization of
the infinite Pascal matrix.
Theorem 3.2. The factorization of the infinite
Pascal matrix is
.
Proof. The generating function from the first
column of matrix is
11
1
⋯
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1
⋯
⋯
⋯⋯
⋯
1
1
1
1⋯
1
⋯
.
From the Riordan matrix, we get
as
follows
2
32
⋯
2
3
⋯
2
3
⋯⋯
2
3
⋯
⋯
.
From definition of the Riordan array, column
generating function is
.
Thus, has the following form
,
1
⋯
1 ,
1.
From the definitions of infinite Pascal matrix and
the infinite Tribonacci polynomials matrix, the
Riordan representations:
,
,
⋯
,.
From the matrix multiplication, the proof is ok.
Secondly, we introduce other factorization of the
Pascal matrix with the Tribonacci polynomials
matrix. Let’s give an infinitive
,
as
follows.
,
⋯
.
We give the infinite by
1000…
1 1 0 0 …
12 2 1 0 …
133 32 3 1 …
⋮⋮⋮⋮…
Now, we introduce the final factorization of the
infinite Pascal matrix.
Theorem 3.3. The factorization of the infinite
Pascal matrix: .
Proof. The proof is similar to Theorem 3.2.
Now, we can give the inverse of Tribonacci
polynomials matrix by helping the definition of the
reverse element of the Riordan group in [11].
Corollary 3.4 The inverse of Tribonacci
polynomial:
1
⋯
,.
4 Conclusion
In this study, new generalized Tribonacci
polynomials have been introduced and studied.
Some combinatorial properties of the Tribonacci
polynomials matrix representations are obtained
with the help of Riordan arrays. In addition, d-
Tribonacci number sequence has been obtained by
considering the Pascal matrix. Based on the Pascal
matrix, two kinds of factors of d-Tribonacci
polynomials were found. Also, infinite
Tribonacci polynomial matrices and the inverses of
these polynomials were found.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
All authors contributed to the study conception and
design. Material preparation, data collection and
analysis were performed by Bahar KULOĞLU,
Engin ÖZKAN. The first draft of the manuscript
was written by Engin ÖZKAN, and all authors
commented on previous versions of the manuscript.
All authors read and approved the final manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funds, grants, or other supports were received.
Conflicts of Interest
All authors certify that they have no affiliations with
or involvement in any organization or entity with
any financial interest or non-financial interest in the
subject matter or materials discussed in this
manuscript.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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