Determinantal Identities of 
-Tetranacci Sequences
PANKAJ
Department of Mathematics, Indira Gandhi University, Meerpur
(Rewari)-122502, Haryana, India
*Corresponding
Author E-mail: pankajarora1242@yahoo.com
Abstract:
In this paper, we find some new determinantal
identities using generalized k-Tetranacci sequences which are defined
as: 
where 
are positive integers with 
The generalized k-Tetranacci sequences are
KEY WORDS: k-Fibonacci sequence, k-Lucas sequence,
k-Tribonacci sequence, k-Tetranacci sequence
2010 MATHEMATICS SUBJECT CLASSIFICATIONS: 11B37, 11B83
1. INTRODUCTION:
The well known Fibonacci sequence has many
interesting properties. The Fibonacci sequence has been generalized in many ways,
some by preserving the initial conditions, and others by preserving the
recurrence relation. Many properties of these types of sequences have been
derived (See [1], [2], [3], [4], [5], [6], [7]). In this paper, we find some
new determinantal identities using generalized k-Tetranacci sequences.
Here, we give definitions of k-Fibonacci, k-Lucas, k-Tribonacci
and k-Tetranacci sequences.
1.1.
Definition:
The k-Fibonacci sequence 
is
defined as,
for 
with
1.2. Definition:
The k-Lucas
sequence 
is
defined as,
for with
1.3. Definition:
The k-Tribonacci
sequence 
is
defined as,
for with
1.4. Definition:
The k-Tetranacci
sequence is
defined as,
for 
with
2.
GENERALIZED
-TETRANACCI
SEQUENCES
Now we define a family of k-Tetranacci
sequences as:
where 
are positive integers with 
The generalized k-Tetranacci sequences are
(1)
(2)
(3)
(4)
(5)
3.
DETERMINANTAL
IDENTITIES OF K- TETRANACCI SEQUENCES:
3.1.
Theorem:
If
are
positive integers with 
then
Proof: Let
Assume 
then by (1),
Now,
Taking 
common from 
and 
respectively, we get
Taking 
common from 
and 
respectively, we get
Applying
we have
Expanding by first row, we get
Applying
we have
Applying
we have
Applying
we have
Put 
and 
we get
The following identities can be proved in a similar
way as in Theorem 3.1.
3.2. Theorem:
If 
are
positive integers with 
then
3.3. Theorem:
If 
are
positive integers with 
then
3.4.
Theorem:
If are
positive integers with then
3.5.
Theorem:
If are
positive integers with then
3.6.
Theorem:
If are
positive integers with then
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On the generating matrices of the k-Fibonacci numbers, Proyecciones
Journal of Mathematics, 32 (4) (2013), 347-357.
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Falcon, and A. Plaza, On the k-Fibonacci numbers, Chaos, Solitons and
Fractals, 5(32) (2007), 1615-1624.
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Falcon, and A. Plaza, The k-Fibonacci hyperbolic functions, Chaos, Solitons and
Fractals, 38(2) (2008), 409-420.
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Falcon, and A. Plaza, The k-Fibonacci sequence and the Pascal 2-triangle,
Chaos, Solitons and Fractals, 33(1) (2007), 38-49.
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Some New Determinantal Identities of 
-Pell
Sequences, Journal
of Combinatorics, Information & System Sciences, 41(4) (2016),
207-213.
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Lather and Manoj Kumar, Stability of k-Tribonacci Functional Equation in
Non-Archimedean Space, International Journal of Computer Applications, 128(14)
(2015), 27-30.
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Sloane, The online encyclopaedia of integer sequences, (2006).
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Received on 30.03.2018
Modified on 13.04.2018
Accepted on 28.04.2018 İA&V
Publications All right reserved
Research J. Science and Tech. 2019; 11(1):09-13.
DOI: 10.5958/2349-2988.2019.00002.0
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