1. Introduction
Besides classical convergence, statistical convergence is an essential advancement that greatly improves the theoretical foundation for sequence spaces in mathematics. Fast [
1] and Steinhaus [
2] independently introduced this concept in the same year. The concept of statistical convergence is technically dependent on the definition of the natural density of subsets of
(the set of all natural numbers). A natural density of a subset
of
is denoted by
and is defined by
in case the limit exists, where
denotes the number of elements of
that are less than or equal to
A sequence
of numbers is called statistically convergent to a number
l if for each
the set
has a natural density zero, i.e.,
This concept has shown its usefulness in various mathematical domains, including summability theory and sequence spaces. Furthermore, statistical convergence has been widely applied in various fields such as trigonometric series, time scales, measure theory, ergodic theory, number theory, cone metric spaces, Banach spaces, and Fourier analysis. Further investigation into this idea and its practical applications can be found in [
3,
4,
5,
6,
7,
8,
9,
10,
11].
Kolk [
12] established the necessary and sufficient Tauberian conditions under which the statistical convergence of a bounded sequence leads to its statistical summability by weighted means. However, this relationship does not generally hold in reverse. Móricz and Orhan [
13] later identified Tauberian conditions that ensure the converse is true for sequences of real and complex numbers. Subsequently, Jena et al. [
14] extended this work by proving Tauberian theorems related to the Cesáro summability of double sequences of fuzzy numbers.
In [
15], the author introduced the difference sequence spaces
and
as follows:
and
where
and the notations
and
denote the spaces of all bounded, convergent, and null sequences, respectively.
In the bottom row, the numbers are called Fibonacci numbers, and the number sequence
is the Fibonacci sequence [
16], and it is denoted by
or
Mathematicians continue to be attracted by the Fibonacci sequence, which is one of the most well-known number sequences in the world. This sequence is extremely valuable and important for mathematicians as a means of broadening their horizons in mathematics. Fibonacci numbers have various fundamental features, such as:
Kara and Basarir [
17] proposed the first use of the Fibonacci sequence in sequence space theory. After that, Kara [
18] generated the Fibonacci difference matrix
utilizing the Fibonacci sequence
and presented new sequence spaces connected to the matrix domain of
Let
be the
Fibonacci number. Then, the infinite matrix
is defined as follows:
By using the same infinite Fibonacci matrix
and a similar technique, Basarir et al. [
19] defined the Fibonacci difference sequence spaces
and
as follows:
and
where
denotes the
-transform of the sequence
defined by
Additional information and applications regarding the utilization of the Fibonacci sequence can be found in [
20,
21,
22,
23,
24,
25,
26].
2. -Fibonacci Statistical Convergence
In this section, we proceed by formally introducing the following new definitions, which are integral to the framework of the study and essential for the comprehensive understanding of the concepts explored in this study.
Definition 1. A sequence of real or complex numbers is called Δ
-Fibonacci convergent (or, -convergent) to a number l ifwhere In this case, we write The class of all -convergent sequences will be represented by that is, Definition 2. A sequence of real or complex numbers is said to be Δ
-Fibonacci statistically convergent (or, -convergent) to a number l if for every the set has zero natural density, i.e.,In this case, we writeThroughout the study, the class of all -convergent sequences will be represented by that is, Definition 3. A sequence of real or complex numbers is said to be strongly Δ
-Fibonacci summable (or, strongly -summable) to a number l ifIn this case, we write or Throughout the study, the class of all strongly -summable sequences will be represented by that is, Theorem 1. If a sequence of real or complex numbers is -convergent, then it is -convergent, that is,
Proof. The proof is straightforward and it is omitted. □
Remark 1. The converse of Theorem 1 is not correct, in general. It is illustrated in the following example.
Example 1. Let us consider the sequence , such thatGiven any Then, for each we haveThis implies thatSo, is -convergent to 0. However, is not -convergent to As a result, Theorem 2. Let be a sequence of real or complex numbers.
If is -convergent to a number then there exists , such that and
is -convergent to a number l if and only if there exists a sequence that is -convergent to l and
Proof. Part (1). Suppose that
is
-convergent to
We need to show that there exists
, such that
and
For this, let us take
for
As
is
-convergent to
we have
It is clear that
for each
We only need to prove the case where some of the
s are non-empty. Assume that
Take
and
such that
and
for all
As a result, we get
with
and
for all
Now, consider a set
and take
Then,
for some
Indeed, let
Then,
Obviously, there exists
, such that
and so
This means that
So, we obtain
for all
Thus,
Next, to show that
Let
be given. Then, we can choose
, such that
For
and
there exists
with
and this implies that
So, we may write
Hence,
Part (2). Suppose that
is
-convergent to a number
Then, by Part (1), there exists
such that
and
Now, define the sequence
, such that
So,
Since
the set
is finite for every
So, there exists
, such that
for all
Thus,
is
-convergent to
To prove
Since
and
then
Conversely, suppose that there exists a sequence
that is
-convergent to l and
Then, for each
we have
As
is
-convergent to
then it is
-convergent to l by Theorem 1. So, the latter set contains a fixed number of integers, let us say
Thus, we may write
This implies that
Therefore,
is
-convergent to
This fulfills the proof. □
Theorem 3. If a sequence is -convergent, then its -limit is unique.
Proof. Suppose that
and
Then, for any
and
Let us take
Then,
So, that
Thus, for any
we may write
As
was arbitrary, we obtain
that is,
□
Theorem 4. Every strongly -summable sequence is -convergent, that is,
Proof. Suppose that
is strongly
-summable to
Then, for every
By taking the limits as
on both sides in the above inequality, we find that
is
-convergent to
Thus,
□
Remark 2. The converse of Theorem 4 is not correct, in general. This can be shown in the following example.
Example 2. Recall the sequence defined in Example 1. Then, we may write where denotes the integral part of the number As as , then Thus, is not strongly -summable to However, is -convergent to 0 as shown in Example 1. As a result, we have
Theorem 5. If a sequence of real or complex numbers is -convergent to l and , such that then is strongly -summable to
Proof. Suppose that
is
-convergent to l and
, such that
Then, for every
we may write
By taking the limits as
on both sides in the above inequality, we obtain that
is strongly
-summable to
□
From Theorem 4 and Theorem 5, we obtain the following result.
Corollary 1. Let be a sequence of real or complex numbers. Then, is strongly -summable to a number l, if and only if it is -convergent to l and , such that
3. Tauberian Conditions
Let us define the first Fibonacci arithmetic means
of a sequence
by setting
Definition 4. A sequence of real or complex numbers is called Δ
-Fibonacci statistically summable (or, -summable) to l if Now, we are ready to present the Tauberian conditions for sequences of real and complex numbers. First, we begin by establishing one-sided Tauberian conditions for sequences of real numbers.
Theorem 6. Let be a sequence of real numbers that is Δ-Fibonacci statistically summable to a finite limit. Then, is -convergent to the same limit, if and only if for each and where denotes the integral part of in symbol
We need the following remarks and results in order to prove Theorem 6.
Remark 3. If conditions (1) and (2) (or equivalently, conditions (2)–(4)) hold, then for all we necessarily haveand for all we have Remark 4. It is possible to modify the proof of Theorem 6 to maintain its validity if conditions (3) and (4) are replaced for the following ones.and Lemma 1. Let and be two sequences of real or complex numbers.
If and then
If c is a constant, then
Proof. The proof is straightforward and it is omitted. □
Lemma 2. If a sequence is Δ
-Fibonacci statistically summable to a number then for each Proof. Let
be given. In case
we have
This implies that
By taking the limits as
on both sides of the inequality (
8), we obtain
In case
According to our claim, the same term
cannot occur more than
times in the sequence
It turns out that if for some integers K and
we have
this implies that
So that
and
that is,
Consequently,
provided N is large enough in the sense that
By taking the limits as
on both sides of the inequality (
9), we obtain that
□
Lemma 3. If a sequence is Δ
-Fibonacci statistically summable to a number then for each and for each Proof. Let
Then, by the definition of
and using the fact
for
then we have
Now, (
10) follows from (
2) and Lemmas 1 and 2, and the fact that for large enough
For
We use this equality this time around:
provided
and n is the large enough in the sense that
and the following inequality for
□
Now, we provide two-sided Tauberian conditions for sequences of complex numbers.
Theorem 7. Let be a sequence of complex numbers that is Δ-Fibonacci statistically summable to a finite limit. Then, is -convergent to the same limit if and only if for each or Now, we are going to provide the proofs of Theorems 6 and 7.
Proof Theorem 6.
Suppose that (
1) and (
2) are satisfied. Then, Lemmas 1 and (3) yields (
5) for
and (
6) for
Conversely, assume that (
2)–(
4) hold. To prove that (
1) holds. We will prove that
Now, for case
it follows from (
12) that
So, for any
Choose any
By (
3), there exists
, such that
By using Lemmas 1 and 2 and (
13), we have
From (
20)–(
22), we have
This means that
Next, for case
By (
14), we may write
That is,
Similarly, by using Lemmas 1 and 2, (
4) and (
15), we obtain
Combining (
23) and (
25), we obtain
This proves (
18). By Lemma 1, we conclude (
1) from (
2) and (
18). □
Proof of Theorem 7.
Suppose that (
1) and (
2) are satisfied. Then, Lemmas 1 and 3 give (
5) for all
and (
6) for all
Conversely, suppose that (
2) and one of (
16) or (
17) are satisfied. We shall show that (
1) holds. For this, it is enough to prove (
18). Let
be given. In case
by (
19), we get
In case
by (
24), we obtain
Choose any
By (
16), there exists
, such that
Or, by (
17), there exists
, such that
From (
26) and (
27), as well as Lemmas 1 and 2, in either case, we obtain
This fulfills (
18). And, Lemma 1 allows us to deduce (
1) from (
2) and (
18). □
4. Conclusions and Suggestions for Further Studies
In the context of sequences, several mathematicians have investigated the principle of statistical convergence. In the current research, the notions of -Fibonacci convergence, -Fibonacci boundedness, -Fibonacci statistical convergence, strong -Fibonacci summability, and -Fibonacci statistical summability of sequences of real and complex numbers have been established. Also, we have obtained one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.
This research paper will provide significant information for future studies in related domains, in addition to researchers performing relevant research in such fields, for instance:
The results and definitions of our study can be used in a
q-calculus, which has applications across various fields of science, such as quantum mechanics, quantum field theory, mathematical physics, number theory, computer science, and discrete mathematics. It can offer a significant generalization of the previously established concepts of
q-statistical convergence,
q-statistical summability and
q-strong Cesàro summability, as presented in [
27]. By extending these foundational definitions, one can develop a more comprehensive framework that incorporates
-Fibonacci statistical convergence, strong
-Fibonacci summability, and
-Fibonacci statistical summability, thereby enriching the theoretical landscape and providing a broader set of tools for analyzing the sequences and series within the context of
q-calculus and its applications.
The concepts of
-Fibonacci statistical convergence with respect to modulus functions and strong
-Fibonacci summability with respect to modulus functions can be introduced as a new research paper (for more details on modulus function, see [
28,
29]).