On -Chebsyhev Sequence
ENGIN OZKAN¹, HAKAN AKKUS²
¹Department of Mathematics,
Faculty of Sciences Arts,
Erzincan Binali Yıldırım University,
Erzincan,
TURKEY
²Department of Mathematics,
Graduate School of Natural and Applied Sciences,
Erzincan Binali Yıldırım University,
Erzincan,
TURKEY
Abstract: - In this study, firstly the -Chebsyhevsequence is defined, and some terms of this sequence are
given. Then, the relations between the terms of the -Chebsyhev sequence are presented and the generating
function of this sequence is obtained. In addition, the Catalan transformation of the sequence is given and the
generating function of the Catalan -Chebsyhev sequence is obtained. Finally, the Hankel transform is applied
to the Catalan -Chebsyhev transform.
Key-Words: - Fibonacci sequences, -Chebsyhev sequences, Catalan Transformation, Cassini Identity, Binet
Formula
Received: November 25, 2022. Revised: May 19, 2023. Accepted: June 12, 2023. Published: July 14, 2023.
1 Introduction
Many studies have been done on the Chebsyhev,
Fibonacci, and Lucas sequence (see for details, [1],
[2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12],
[13]). In [14], Chebsyhev polynomials of the first
and second kind are defined as follows,
respectively,
with
,
, with
.
In [15], the relationship between Chebsyhev
polynomials and matrices was examined. In [16],
the authors defined the -Fibonacci sequence and
gave the properties related to the -Fiboonacci
sequence. Then the -Pell, -Jacobsthal, -Pell-
Lucas, Modified -Pell and -Jacobsthal-Lucas
numbers were defined in [17], [18], [19],
respectively,
, with
,
, with
,
, with
,
, with
and
, with
.
In addition, in [20], [21], -Fibonacci, Pell
numbers, Pell-Lucas numbers, Jacobsthal numbers,
and Jacobsthal-Lucas numbers sequences which are
a new generalization of the Fibonacci sequence
were defined, and their properties were found. In
[22], [23], [24], Hankel and Catalan's
transformations were defined and some of their
properties were brought to the literature.
In Chapter 2, we define the -Chebsyhev sequence,
then give the characteristic equation, the Binet
formula, and some properties of the sequence. We
also show the relationship between the positive and
negative terms of the sequence, get a relationship
between three consecutive terms, and examine the
relationship between the terms in the limit infinite
case. Finally in this chapter, we give the Cassini and
the Honsberger identity for this sequence.
In Chapter 3, the Catalan transformation of the -
Chebsyhev sequence is defined, and some properties
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.56
Engin Ozkan, Hakan Akkus
E-ISSN: 2224-2880
503
Volume 22, 2023
are given. Finally, the Hankel transform for the -
Chebsyhev sequence is calculated.
2 -Chebsyhev Sequence
Let's take a positive real number for the sequence
we are going to define. The -Chebsyhev sequences
are as follows;
with and .
Then, let’s write the characteristic equation for the
sequence, the roots of the characteristic equation,
and the Binet formula.
The characteristic equation of the sequence is
.
The roots of the characteristic equation of the
sequence are as follows;
and
where , ,
.
The Binet formula of the defined sequence is as
follows.
=
.
Now let’s give the first elements of -Chebyshev
sequence.
For some n, we give some values of the -
Chebyshev sequence in Table 1.
Table 1. -Chebyshev sequence for
Theorem The Binet formula of the -Chebsyhev
sequence is as follows;
.
Proof The general solution of a sequence is
.
Here, the scalars and can be obtained by
substituting the initial conditions. It is obtained by
solving the given system of equations. For it
is and for it is . Thus
and
are obtained. From here
=
.
Theorem Let be the positive and big root of the
characteristic equation of -Chebsyhev sequence.
.
Proof
()
.
Theorem The following relation is satisfied.
.
Proof If the Binet formula of the sequence -
Chebsyhev is used
=
.
=
=
( )
.
Thus
= .
Theorem (Cassini Identity)
.-
= .
Proof From the Binet formula of the sequence -
Chebsyhev, then we have
.-
=
)(
)
=
=
+
–
=
+
= .
Theorem (Honsberger Identity)
Proof By the Binet formula of the sequence -
Chebsyhev, we obtain
-(+
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.56
Engin Ozkan, Hakan Akkus
E-ISSN: 2224-2880
504
Volume 22, 2023
.
Theorem There is the following relationship
between the squares of consecutive terms of the
sequence .
.
Proof
.
Theorem (Generating Function)
Proof The following equations are written for the -
Chebsyhev sequence,
.
Thus
.
3 Catalan Number
In [23], [24], the Catalan numbers where is a
positive integer are as follows.
or
and the tensile function is given as
for some natural numbers . Catalan numbers are
shaped.
3.1 Catalan Transformation for the -
Chebyshev Sequences
Using the Catalan transformation, we define the
Catalan transformation of the -Chebsyhev
sequences {} as followed.
=
with = 0.
Now we can give the Catalan transformation of the
first elements of the -Chebsyhev sequence.
=
=
Some values of the Catalan transformation of the -
Chebsyhev sequence is given Table 2.
Table 2. Catalan transformation of the -Chebsyhev
sequence
We can show { as the matrix and the
product of the lower triangular matrix as
So,
=
6
48
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.56
Engin Ozkan, Hakan Akkus
E-ISSN: 2224-2880
505
Volume 22, 2023
3.2 The Generating Function Catalan -
Chebsyhev Sequence
The generating function of -Chebsyhev and
Catalan sequences are as follows, respectively,
and
.
Thus, the following equations are written for the
generating function of the Catalan -Chebsyhev
sequence,
.
3.3 Hankel Transform of the Catalan -
Chebsyhev Sequences
In [22], [23], let the set of the terms of a sequence
be . The Hankel transform of
the terms of this sequence is defined as follows.
(5)
For example, the Hankel matrix of the 3rd Lucas
sequence,
.
The determinant of this matrix .
If we apply Hankel’s work to the Catalan -
Chebsyhev sequence, we finally get;
=
=
.
4 Conclusions
In this paper, we first defined the k- Chebsyhev. We
then gave the main features of this sequence. We
also examined the relationships between the terms
of this sequence. Finally, we introduced the Catalan
and Hankel transformation of the sequence. This
work can be further extended to Horadam numbers
and Mersenne numbers.
References
[1] Pucanović, Z., & Pešović, M., Chebyshev
polynomials and r-circulant matrices. Applied
Mathematics and Computation, 437, (2023)
127521.
[2] Kishore, J., & Verma, V. Some identities
involving Chebyshev polynomials, Fibonacci
polynomials and their derivatives.
[3] Özkan, E., & Tastan, M., (2021). On a new
family of gauss k-Lucas numbers and their
polynomials. Asian-European Journal Of
Mathematics, vol.14, no.6.
[4] Özkan, E., Taştan, M., Aydoğdu, A. ‘’3-
Fibonacci Polynomials in the Family of
Fibonacci Numbers’’, Erzincan University
Journal of the Institute of Science and
Technology, 12(2), (2019), 926-933.
[5] Özkan, E., Uysal, M., & Godase, A.,
Hyperbolic k-Jacobsthal and k-Jacobsthal-
Lucas Quaternions. Indian Journal of Pure and
Applied Mathematics, (2022), vol.53, no.4,
956-967.
[6] Özkan, E., Uysal, M., "d-Gaussian Fibonacci,
d-Gaussian Lucas Polynomials and Their
Matrix Representations, "Ukranian
Mathematical Journal, vol.75,(2023), pp.1-17.
[7] Koshy, T., ‘’Fibonacci and Lucas Numbers
with Applications’’. New York, NY:, Wiley-
Interscience Publication, 2001.
[8] Uysal, M., Kumarı, M., Kuloğlu, B., Prasad,
K., & Özkan, E., On the Hyperbolıc k-
Mersenne and k-Mersenne-Lucas Octonions.
Kragujevac Journal Of Mathematics, (2025),
vol.49, no.5, 765-779.
[9] Bród, D., Szynal-Liana, A., & Włoch, I.
(2022). One-parameter generalization of the
bihyperbolic Jacobsthal numbers. Asian-
European Journal of Mathematics, 2350075.
[10] Taştan, M., Yılmaz, N., & Özkan, E., (2022).
A new family of Gauss (k,t)-Horadam
numbers. Asian-European Journal of
Mathematics, vol.15, no.12, 22502254.
[11] Kuloğlu, B., Özkan, E., & Marin, M.,
Fibonacci and Lucas Polynomials in n-gon.
Analele Stiintifice Ale Universitatii Ovidius
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.56
Engin Ozkan, Hakan Akkus
E-ISSN: 2224-2880
506
Volume 22, 2023
Constante, Seria Matematica, (2023), vol.31,
no.2, 127-140.
[12] Özkan, E., & Altun, İ., ‘’Generalized Lucas
polynomials and relationships between the
Fibonacci polynomials and Lucas
polynomials. Communications In Algebra,
(2019) vol.47, no.10, 4020-4030.
[13] Özkan, E., Tastan, M., On Gauss Fibonacci
polynomials, on Gauss Lucas polynomials and
their applications. Communications In
Algebra, (2020), vol.48, no.3, 952- 960.
[14] Abchiche, Mourad, and Hacéne Belbachir.
"Generalized chebyshev polynomials."
Discussiones Mathematicae-General Algebra
and Applications 38.1 (2018): 79-90.
[15] İleri, A., & Küçüksakallı, Ö. On the Jacobian
Matrices of Generalized Chebyshev
Polynomials, (2022), arXiv preprint
arXiv:2212.08381.
[16] Falc´on, S., and Plaza, A. ‘’On the Fibonacci
k-numbers’’, Chaos Solitons Fractals, 32(5),
(2007), 1615–1624.
[17] Taştan, M.,Özkan, E. ‘’Catalan Transform of
the k−Jacobsthal Sequence’’, Electronic
Journal of Mathematical Analysis and
Applications, 8(2), (2020), 70-74.
[18] Taştan M., Özkan E. “Catalan Transform of
the k-Pell, k-Pell-Lucas and Modified k-Pell
Sequence’’, Notes on Number Theory and
Discrete Mathematics, 27(1), (2021), 198–
207.
[19] Akkuş, H., Üregen, N., and Özkan E. “A New
Approach to −Jacobsthal
LucasSequences’’,Sakarya University Journal
of Science, 25(4), (2021), 969-973.
[20] Falcon, S. ‘’Catalan Transform of the k-
Fibonacci sequence’’. Commun. Korean
Math. Soc., 28(4), (2013), 28(4), 827–832.
[21] Çelik, S., Durukan, İ. and Özkan, E. (2021)
“New Recurrences on Pell Numbers, Pell
Lucas Numbers, Jacobsthal Numbers, And
Jacobsthal Lucas Numbers, Chaos, Solitons
and Fractals’’ Journal of Science and Arts,
150, (2021), 111173.
[22] Layman, J. W. ‘’The Hankel transform and
some of its properties’’. J. Integer Seq., 4(1),
(2001), 1-11.
[23] Rajkovi´c, P. M., Petkovi´c, M. D. and Barry,
P., ‘’The Hankel transform of the sum of
consecutive generalized Catalan numbers.
Integral Transforms Spec. Funct’’, 18(4),
(2007), 285–296.
[24] Barry, P. ‘’A Catalan transform and related
transformations on integer sequences’’.
J.Integer Seq., 8(4), (2005), 1-24.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Engin ÖZKAN carried out the introduction and the
main result of the article.
-Hakan AKKUŞ has improved Chapter 2 and
Chapter 3. All authors read and approved the final
manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This study did not receive any funding in any form.
Conflict of Interest
The authors have no conflict of interest to declare.
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
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.56
Engin Ozkan, Hakan Akkus
E-ISSN: 2224-2880
507
Volume 22, 2023
