In this paper, we introduce a q-analog of the bi-periodic Lucas sequence, called as the q-bi-periodic Lucas sequence, and give some identities related to the q-bi-periodic Fibonacci and Lucas ...sequences. Also, we give a matrix representation for the q-bi-periodic Fibonacci sequence which allow us to obtain several properties of this sequence in a simple way. Moreover, by using the explicit formulas for the q-bi-periodic Fibonacci and Lucas sequences, we introduce q-analogs of the bi-periodic incomplete Fibonacci and Lucas sequences and give a relation between them.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid ...numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.
In this paper, we consider a generalization of Horadam sequence
{
w
n
}
which is defined by the recurrence relation
w
n
=
χ
(
n
)
w
n
−
1
+
c
w
n
−
2
, where
χ
(
n
)
=
a
if
n
is even,
χ
(
n
)
=
b
if
...n
is odd with arbitrary initial conditions
w
0
,
w
1
and nonzero real numbers
a
,
b
and
c
. As a special case, by taking the initial conditions 0, 1 and 2,
b
we define the sequences
{
u
n
}
and
{
v
n
}
, respectively. The main purpose of this study is to derive some basic properties of the sequences
{
u
n
}
,
{
v
n
}
and
{
w
n
}
by using a matrix approach.
It is well-known that the ratios of successive terms of Fibonacci numbers {Fn+1 /Fn} o converge to the golden ratio (1+p5/ 2} , so it is natural to ask if analogous results exist for the ...generalizations of the Fibonacci sequence. In this paper, we consider the generalization of the Fibonacci sequence, which is called Fibonacci-like conditional sequences and we investigate the convergence properties of this sequences.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
Some results on Horadam quaternions Tan, Elif; Leung, Ho-Hon
Chaos, solitons and fractals,
September 2020, 2020-09-00, Letnik:
138
Journal Article
Recenzirano
•Give a simple and practical way to obtain many properties of Fibonacci like quaternions by using the matrix method.•Give oportunity to obtain many matrix identities by choosing appropriate initial ...values in our matrix formula.•Prevent mathematical errors that may occur in quaternion multiplication.•We give non-commutative analogs of the binomial sum identities, CatalanLike identities, Cassini-Like identities and d’Ocagne-Like identities, etc.•Derive more general results for Horadam quaternions by using the commutator bracket.
In this work, we present some matrix representations associated with the Horadam quaternions which are defined by Wn=wn+wn+1i+wn+2j+wn+3k, where the components are taken from the Horadam sequence {wn}. We derive many identities related to them by using the matrix technique which is more practical. Since various well-known Fibonacci-type quaternion matrices are special cases of Horadam quaternion matrices, we have a unified way of dealing with many special quaternion sequences in the literature. As an application, we derive some binomial-sum identities.
The hybrid numbers were introduced by Ozdemir
9
as a new generalization of complex, dual, and hyperbolic numbers. A hybrid number is defined by
k
=
a
+
b
i
+
c
ϵ
+
d
h
, where
a
,
b
,
c
,
d
are ...real numbers and
i
,
ϵ
,
h
are operators such that
i
2
=
-
1
,
ϵ
2
=
0
,
h
2
=
1
and
i
h
=
-
h
i
=
ϵ
+
i
. This work is intended as an attempt to introduce the bi-periodic Horadam hybrid numbers which generalize the classical Horadam hybrid numbers. We give the generating function, the Binet formula, and some basic properties of these new hybrid numbers. Also, we investigate some relationships between generalized bi-periodic Fibonacci hybrid numbers and generalized bi-periodic Lucas hybrid numbers.
In this study, we introduce a new class of generalized quaternions whose components are dual-generalized complex Horadam numbers. We investigate some algebraic properties of them.
Motivated by the our recent work in Tan et al., 2016, related to the bi-periodic Fibonacci quaternions, here we introduce the bi-periodic Lucas quaternions that gives the Lucas quaternions as a ...special case. We give the generating function and the Binet formula for these quaternions. Also, we give the relationships between bi-periodic Fibonacci quaternions and bi-periodic Lucas quaternions.
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