A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

Abstract

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called $\Delta$-Fibonacci statistical convergence, strong $\Delta$-Fibonacci summability, and $\Delta$-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

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 $\mathbb{N}$ (the set of all natural numbers). A natural density of a subset $\Omega$ of $\mathbb{N}$ is denoted by $\delta \left(\Omega \right)$ and is defined by

$$\delta \left(\Omega \right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{i\le n:i\in \Omega \right\}\right|,$$

in case the limit exists, where $\left|\left\{i\le n:i\in \Omega \right\}\right|$ denotes the number of elements of $\Omega$ that are less than or equal to $n.$

A sequence $\left(x_k\right)$ of numbers is called statistically convergent to a number l if for each $\epsilon >0,$ the set $\left\{k\in \mathbb{N}:\left|x_k-l\right|\ge \epsilon \right\}$ has a natural density zero, i.e.,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le \mathbb{N}:\left|x_k-l\right|\ge \epsilon \right\}\right|=0.$$

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 ${\ell }_{\infty }\left(\Delta \right),$$c\left(\Delta \right),$ and $c_0\left(\Delta \right)$ as follows:

$${\ell }_{\infty }\left(\Delta \right)=\left\{x=\left(x_k\right):\Delta x\in {\ell }_{\infty }\right\},$$

$$c\left(\Delta \right)=\left\{x=\left(x_k\right):\Delta x\in c\right\},$$

and

$$c_0\left(\Delta \right)=\left\{x=\left(x_k\right):\Delta x\in c_0\right\},$$

where $\Delta x=\left(\Delta x_k\right)=\left(x_k-x_{k+1}\right)$ and the notations ${\ell }_{\infty },$$c,$ and $c_0$ 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

$$\left(1,1,2,3,5,8,13,21,34,55,89,144,\dots \right)$$

is the Fibonacci sequence [16], and it is denoted by ${\left(f_n\right)}_{n=0}^{\infty }$ or $\left(f_n\right).$ 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:

$$\underset{n\to \infty }{lim}{\displaystyle \frac{f_{n+1}}{f_n}}={\displaystyle \frac{1+\sqrt{5}}{2}}=\rho \left(\mathrm{Golden}\mathrm{ratio}\right),$$

$$\sum _k{\displaystyle \frac{1}{f_k}}\mathrm{converges},$$

$$\sum _{k=0}^n{f}_k=f_{n+2}-1\mathrm{for}n\in \mathbb{N},$$

$$f_{n-1}{f}_{n+1}-f_n^2={\left(-1\right)}^{n+1}\mathrm{for}\mathrm{all}n\ge 1\left(\mathrm{Cassinis}\mathrm{formula}\right).$$

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 $\widehat{F}$ utilizing the Fibonacci sequence $\left(f_n\right)$ and presented new sequence spaces connected to the matrix domain of $\widehat{F}.$ Let $f_n$ be the $n\mathrm{th}$ Fibonacci number. Then, the infinite matrix $\widehat{F}=\left(\widehat{f_{nk}}\right)$ is defined as follows:

$$\widehat{f_{nk}}=\left\{\begin{array}{cc}-{\displaystyle \frac{f_{n+1}}{f_n}},\hfill & k=n-1,\hfill \\ {\displaystyle \frac{f_n}{f_{n+1}}},\hfill & k=n,\hfill \\ 0,\hfill & 0\le k<n-1\mathrm{or}k>n.\hfill \end{array}\right.$$

By using the same infinite Fibonacci matrix $\widehat{F}$ and a similar technique, Basarir et al. [19] defined the Fibonacci difference sequence spaces $c_0\left(\widehat{F}\right)$ and $c\left(\widehat{F}\right)$ as follows:

$$c_0\left(\widehat{F}\right)=\left\{x=\left(x_k\right):\underset{k\to \infty }{lim}\widehat{F_k}\left(x\right)=0\right\}$$

and

$$c\left(\widehat{F}\right)=\left\{x=\left(x_k\right):\underset{k\to \infty }{lim}\widehat{F_k}\left(x\right)=l\mathrm{for}\mathrm{some}\mathrm{number}l\right\},$$

where $\widehat{F_k}\left(x\right)$ denotes the $\widehat{F}$-transform of the sequence $\left(x_k\right),$ defined by

$$\widehat{F_k}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{f_0}{f_1}}{x}_0=x_0,\hfill & k=0,\\ \hfill & \\ {\displaystyle \frac{f_k}{f_{k+1}}}{x}_k-{\displaystyle \frac{f_{k+1}}{f_k}}{x}_{k-1},\hfill & k\ge 1.\end{array}\right.$$

Additional information and applications regarding the utilization of the Fibonacci sequence can be found in [20,21,22,23,24,25,26].

2. $\Delta$-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 $\left(x_k\right)$ of real or complex numbers is called Δ-Fibonacci convergent (or, $\Delta \widehat{F}$-convergent) to a number l if

$$\underset{k\to \infty }{lim}\Delta \widehat{F_k}\left(x\right)=l,$$

where $\Delta \widehat{F_k}\left(x\right)=\widehat{F_k}\left(x\right)-\widehat{F_{k+1}}\left(x\right).$ In this case, we write $\Delta \widehat{F}-lim{x}_k=l.$ The class of all $\Delta \widehat{F}$-convergent sequences will be represented by $c\left[\Delta \widehat{F}\right],$ that is,

$$c\left[\Delta \widehat{F}\right]=\left\{\left(x_k\right):\Delta \widehat{F}-lim{x}_k=l\mathit{for}\mathit{some}\mathit{number}l\right\}.$$

Definition 2.

A sequence $\left(x_k\right)$ of real or complex numbers is said to be Δ-Fibonacci statistically convergent (or, $\left(\Delta \widehat{F},S\right)$-convergent) to a number l if for every $\epsilon >0,$ the set $\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}$ has zero natural density, i.e.,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|=0.$$

In this case, we write

$$St-lim\Delta \widehat{F_k}\left(x\right)=l$$

(1)

Throughout the study, the class of all $\left(\Delta \widehat{F},S\right)$-convergent sequences will be represented by $S\left[\Delta \widehat{F}\right],$ that is,

$$S\left[\Delta \widehat{F}\right]=\left\{\left(x_k\right):\forall \epsilon >0,\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|=0\mathit{for}\mathit{some}\mathit{number}l\right\}.$$

Definition 3.

A sequence $\left(x_k\right)$ of real or complex numbers is said to be strongly Δ-Fibonacci summable (or, strongly $\left(\Delta \widehat{F},N\right)$-summable) to a number l if

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|=0.$$

In this case, we write $\left(\Delta \widehat{F},N\right)-lim{x}_k=l$ or $x_k\to l\left(\Delta \widehat{F},N\right).$ Throughout the study, the class of all strongly $\left(\Delta \widehat{F},N\right)$-summable sequences will be represented by $N\left[\Delta \widehat{F}\right],$ that is,

$$N\left[\Delta \widehat{F}\right]=\left\{\left(x_k\right):\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|=0\mathit{for}\mathit{some}\mathit{number}l\right\}.$$

Theorem 1.

If a sequence $\left(x_k\right)$ of real or complex numbers is $\Delta \widehat{F}$-convergent, then it is $\left(\Delta \widehat{F},S\right)$-convergent, that is, $c\left[\Delta \widehat{F}\right]\subset S\left[\Delta \widehat{F}\right].$

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 $\left(x_k\right)$, such that

$$\Delta \widehat{F_k}\left(x\right)=\left\{\begin{array}{cc}k\hfill & \mathrm{if}k=n^3\hfill \\ \hfill \\ 0\hfill & \mathrm{if}k\ne n^3\hfill \end{array}\right.n\in \mathbb{N}.$$

Given any $\epsilon >0.$ Then, for each $n\in \mathbb{N},$ we have

$$\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)\right|\ge \epsilon \right\}\right|\le \sqrt[3]{n}.$$

This implies that

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)\right|\ge \epsilon \right\}\right|\le \underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sqrt[3]{n}=0.$$

So, $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to 0. However, $\left(x_k\right)$ is not $\Delta \widehat{F}$-convergent to $0.$ As a result, $c\left[\Delta \widehat{F}\right]⫋S\left[\Delta \widehat{F}\right].$

Theorem 2.

Let $\left(x_k\right)$ be a sequence of real or complex numbers.

• If $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to a number $l,$ then there exists $\Omega \subset \mathbb{N}$, such that $\delta \left(\Omega \right)=0$ and $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l.$
• $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to a number l if and only if there exists a sequence $\left(y_k\right)$ that is $\Delta \widehat{F}$-convergent to l and $\delta \left(\left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\right)=0.$

Proof. 

Part (1). Suppose that $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to $l.$ We need to show that there exists $\Omega \subset \mathbb{N}$, such that $\delta \left(\Omega \right)=0$ and $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l.$ For this, let us take ${\Omega }^i=\left\{k\in \mathbb{N}:\left|\widehat{F_k}\left(x\right)-l\right|>{\displaystyle \frac{1}{i}}\right\}$ for $i\in \mathbb{N}.$ As $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to $l,$ we have $\delta \left({\Omega }^i\right)=0.$ It is clear that ${\Omega }^i\subset {\Omega }^{i+1}$ for each $i\in \mathbb{N}.$ We only need to prove the case where some of the ${{\Omega }^i}^{\prime }$s are non-empty. Assume that ${\Omega }^1\ne \varphi .$ Take $a_1\in {\Omega }^1$ and $a_2\in {\Theta }^2$ such that $a_2>a_1$ and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|>{\displaystyle \frac{1}{2}}\right\}\right|<{\displaystyle \frac{1}{2}}$$

for all $n\ge a_2.$ As a result, we get $a_1<a_2<a_3<\dots$ with $a_i\in {\Omega }^i$ and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|>{\displaystyle \frac{1}{i}}\right\}\right|<{\displaystyle \frac{1}{i}}$$

for all $n\ge a_i.$ Now, consider a set $\Omega =\left(\left[a_i,a_{i+1}\right)\cap {\Omega }^i\right)$ and take ${\Omega }^i\left(n\right)=$$\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|>{\displaystyle \frac{1}{i}}\right\}.$ Then, $\Omega \left(n\right)\subset {\Omega }^i\left(n\right)$ for some $i.$ Indeed, let $z\in \Omega \left(n\right).$ Then, $z\ge a_1.$ Obviously, there exists $i\in \mathbb{N}$, such that $a_i\le z<a_{i+1}$ and so $z\in {\Omega }^i\left(n\right).$ This means that $\Omega \left(n\right)\subset {\Omega }^i\left(n\right).$ So, we obtain

$${\displaystyle \frac{\left|\Omega \left(n\right)\right|}{n}}\le {\displaystyle \frac{\left|{\Omega }^i\left(n\right)\right|}{n}}\le {\displaystyle \frac{1}{i}}$$

for all $n\ge a_i.$ Thus, $\delta \left(\Omega \right)=0.$

Next, to show that $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l.$ Let $\epsilon >0$ be given. Then, we can choose $i\in \mathbb{N}$, such that ${\displaystyle \frac{1}{i}}<\epsilon .$ For $k\in \mathbb{N}\setminus \Omega$ and $k\ge a_i,$ there exists $t\ge i$ with $i_t\le k\le i_{t+1}$ and this implies that $k\notin {\Omega }^t.$ So, we may write

$$\left|\Delta \widehat{F_k}\left(x\right)-l\right|<{\displaystyle \frac{1}{t}}\le {\displaystyle \frac{1}{i}}<\epsilon .$$

Hence, $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l.$

Part (2). Suppose that $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to a number $l.$ Then, by Part (1), there exists $\Omega \subset \mathbb{N}$ such that $\delta \left(\Omega \right)=0$ and $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l.$ Now, define the sequence $\left(y_k\right)$, such that

$$\Delta \widehat{F_k}\left(y\right)=\left\{\begin{array}{cc}l,\hfill & \mathrm{if}k\in \Omega ,\hfill \\ \hfill & \\ \Delta \widehat{F_k}\left(x\right),\hfill & \mathrm{if}k\in \mathbb{N}\setminus \Omega .\hfill \end{array}\right.$$

So,

$$\Delta \widehat{F_k}\left(y\right)-l=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}k\in \Omega ,\hfill \\ \hfill & \\ \Delta \widehat{F_k}\left(x\right)-l,\hfill & \mathrm{if}k\in \mathbb{N}\setminus \Omega .\hfill \end{array}\right.$$

Since $\underset{k\in \mathbb{N}\setminus \Omega }{lim}\Delta \widehat{F_k}\left(x\right)=l,$ the set $\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(y\right)-l\right|>\epsilon \right\}$ is finite for every $\epsilon >0.$ So, there exists $n_0\in \mathbb{N}$, such that $\left|\Delta \widehat{F_k}\left(y\right)-l\right|<\epsilon$ for all $k>n_0.$ Thus, $\left(y_k\right)$ is $\Delta \widehat{F}$-convergent to $l.$ To prove $\delta \left(\left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\right)=0.$ Since $\left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\subset \Omega$ and $\delta \left(\Omega \right)=0,$ then $\delta \left(\left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\right)=0.$

Conversely, suppose that there exists a sequence $\left(y_k\right)$ that is $\Delta \widehat{F}$-convergent to l and $\delta \left(\left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\right)=0.$ Then, for each $\epsilon >0,$ we have

$$\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\subset \left\{k\in \mathbb{N}:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\cup \left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(y\right)-l\right|>\epsilon \right\}.$$

As $\left(y_k\right)$ is $\Delta \widehat{F}$-convergent to $l,$ then it is $\left(\Delta \widehat{F},S\right)$-convergent to l by Theorem 1. So, the latter set contains a fixed number of integers, let us say $\phi =\phi \left(\epsilon \right).$ Thus, we may write

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|\le \underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\Delta \widehat{F_k}\left(x\right)\ne \Delta \widehat{F_k}\left(y\right)\right\}\right|+\underset{n\to \infty }{lim}{\displaystyle \frac{\phi }{n}}=0.$$

This implies that

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|=0.$$

Therefore, $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to $l.$ This fulfills the proof. □

Theorem 3.

If a sequence $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent, then its $\left(\Delta \widehat{F},S\right)$-limit is unique.

Proof. 

Suppose that $St-lim\Delta \widehat{F_k}\left(x\right)=l_1$ and $St-lim\Delta \widehat{F_k}\left(x\right)=l_2.$ Then, for any $\epsilon >0,$

$$\delta \left(\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l_1\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\right)=0$$

and

$$\delta \left(\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l_2\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\right)=0.$$

Let us take $\Theta \left(\epsilon \right)=\left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l_1\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\cup \left\{k\in \mathbb{N}:\left|\Delta \widehat{F_k}\left(x\right)-l_2\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}.$ Then, $\delta \left(\Theta \left(\epsilon \right)\right)=0.$ So, that $\mathbb{N}\setminus \Theta \left(\epsilon \right)\ne \varphi .$ Thus, for any $k\in \mathbb{N}\setminus \Theta \left(\epsilon \right),$ we may write

$$\left|l_1-l_2\right|\le \left|l_1-\Delta \widehat{F_k}\left(x\right)\right|+\left|\Delta \widehat{F_k}\left(x\right)-l_2\right|<{\displaystyle \frac{\epsilon }{2}}+{\displaystyle \frac{\epsilon }{2}}=\epsilon .$$

As $\epsilon >0$ was arbitrary, we obtain $\left|l_1-l_2\right|=0,$ that is, $l_1=l_2.$ □

Theorem 4.

Every strongly $\left(\Delta \widehat{F},N\right)$-summable sequence is $\left(\Delta \widehat{F},S\right)$-convergent, that is, $N\left[\Delta \widehat{F}\right]\subset S\left[\Delta \widehat{F}\right].$

Proof. 

Suppose that $\left(x_k\right)$ is strongly $\left(\Delta \widehat{F},N\right)$-summable to $l.$ Then, for every $\epsilon >0,$

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|& ={\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ \left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \end{array}}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|+{\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ \left|\Delta \widehat{F_k}\left(x\right)-l\right|<\epsilon \end{array}}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|\hfill \\ \hfill & \ge {\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ \left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \end{array}}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|\hfill \\ \hfill & \ge {\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|·\epsilon .\hfill \end{array}$$

By taking the limits as $n\to \infty$ on both sides in the above inequality, we find that $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to $l.$ Thus, $N\left[\Delta \widehat{F}\right]\subset S\left[\Delta \widehat{F}\right].$ □

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 $\left(x_k\right)$ defined in Example 1. Then, we may write

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-0\right|& ={\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ k=m^3\end{array}}^{n}k+{\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ k\ne m^3\end{array}}^{n}0\hfill \\ \hfill & ={\displaystyle \frac{1}{n}}\left(1^3+2^3+\cdots +q^3\right),\underset{q\in \mathbb{N}}{max{q}^3\le n}\hfill \\ \hfill & ={\displaystyle \frac{1}{n}}{\left[{\displaystyle \frac{q\left(q+1\right)}{2}}\right]}^2\hfill \\ \hfill & \ge {\displaystyle \frac{1}{n}}{\left[{\displaystyle \frac{\left(\left[\sqrt[3]{n}\right]-1\right)\left[\sqrt[3]{n}\right]}{2}}\right]}^2,\hfill \end{array}$$

where $\left[\sqrt[3]{n}\right]$ denotes the integral part of the number $\sqrt[3]{n}.$ As ${\displaystyle \frac{1}{n}}{\left[{\displaystyle \frac{\left(\left[\sqrt[3]{n}\right]-1\right)\left[\sqrt[3]{n}\right]}{2}}\right]}^2\to \infty$ as $n\to \infty$, then

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-0\right|\ne 0.$$

Thus, $\left(x_k\right)$ is not strongly $\left(\Delta \widehat{F},N\right)$-summable to $0.$ However, $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to 0 as shown in Example 1. As a result, we have $N\left[\Delta \widehat{F}\right]⫋S\left[\Delta \widehat{F}\right].$

Theorem 5.

If a sequence $\left(x_k\right)$ of real or complex numbers is $\left(\Delta \widehat{F},S\right)$-convergent to l and $\exists P\in {\mathbb{R}}^{+}$, such that $\left|\Delta \widehat{F_k}\left(x\right)-l\right|\le P\forall k\in \mathbb{N},$ then $\left(x_k\right)$ is strongly $\left(\Delta \widehat{F},N\right)$-summable to $l.$

Proof. 

Suppose that $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to l and $\exists P\in {\mathbb{R}}^{+}$, such that $\left|\Delta \widehat{F_k}\left(x\right)-l\right|\le P\forall k\in \mathbb{N}.$ Then, for every $\epsilon >0,$ we may write

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}\sum _{k=1}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|& ={\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ \left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \end{array}}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|+{\displaystyle \frac{1}{n}}\sum _{\begin{array}{c}k=1\\ \left|\Delta \widehat{F_k}\left(x\right)-l\right|<\epsilon \end{array}}^n\left|\Delta \widehat{F_k}\left(x\right)-l\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|\Delta \widehat{F_k}\left(x\right)-l\right|\ge \epsilon \right\}\right|·P+\epsilon .\hfill \end{array}$$

By taking the limits as $n\to \infty$ on both sides in the above inequality, we obtain that $\left(x_k\right)$ is strongly $\left(\Delta \widehat{F},N\right)$-summable to $l.$ □

From Theorem 4 and Theorem 5, we obtain the following result.

Corollary 1.

Let $\left(x_k\right)$ be a sequence of real or complex numbers. Then, $\left(x_k\right)$ is strongly $\left(\Delta \widehat{F},N\right)$-summable to a number l, if and only if it is $\left(\Delta \widehat{F},S\right)$-convergent to l and $\exists P\in {\mathbb{R}}^{+}$, such that $\left|\Delta \widehat{F_k}\left(x\right)-l\right|\le P\forall k\in \mathbb{N}.$

3. Tauberian Conditions

Let us define the first Fibonacci arithmetic means $\Delta \widehat{F_n}\left(\sigma \right)$ of a sequence $\left(x_k\right)$ by setting

$$\Delta \widehat{F_n}\left(\sigma \right)={\displaystyle \frac{1}{n+1}}\sum _{k=0}^n\Delta \widehat{F_k}\left(x\right),n=0,1,2,\dots .$$

Definition 4.

A sequence $\left(x_k\right)$ of real or complex numbers is called Δ-Fibonacci statistically summable (or, $\left(\Delta \widehat{F},C,1\right)$-summable) to l if

$$St-lim\Delta \widehat{F_n}\left(\sigma \right)=l.$$

(2)

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 $\left(x_k\right)$ be a sequence of real numbers that is Δ-Fibonacci statistically summable $\left(\Delta \widehat{F},C,1\right)$ to a finite limit. Then, $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to the same limit, if and only if for each $\epsilon >0,$

$$\underset{\lambda >1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\le -\epsilon \right\}\right|=0$$

(3)

and

$$\underset{0<\lambda <1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)\le -\epsilon \right\}\right|=0,$$

(4)

where ${\lambda }_n$ denotes the integral part of $\lambda n,$ in symbol ${\lambda }_n=\left[\lambda n\right].$

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 $\lambda >1,$ we necessarily have

$$St-lim{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)=0;$$

(5)

and for all $0<\lambda <1,$ we have

$$St-lim{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)=0.$$

(6)

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.

$$\underset{\lambda >1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\ge \epsilon \right\}\right|=0$$

and

$$\underset{0<\lambda <1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)\ge \epsilon \right\}\right|=0.$$

Lemma 1.

Let $\left(x_k\right)$ and $\left(y_k\right)$ be two sequences of real or complex numbers.

• If $St-lim\Delta \widehat{F_k}\left(x\right)=l_1$ and $St-lim\Delta \widehat{F_k}\left(y\right)=l_2,$ then $St-lim\left(\Delta \widehat{F_k}\left(x\right)+\Delta \widehat{F_k}\left(y\right)\right)=l_1+l_2.$
• If c is a constant, then $St-limc\Delta \widehat{F_k}\left(x\right)=c{l}_1.$

Proof. 

The proof is straightforward and it is omitted. □

Lemma 2.

If a sequence $\left(x_k\right)$ is Δ-Fibonacci statistically summable $\left(\Delta \widehat{F},C,1\right)$ to a number $l,$ then for each $\lambda >0,$

$$St-lim\widehat{F_{{\lambda }_n}}\left(\sigma \right)=l,\mathrm{where}{\lambda }_n=\left[\lambda n\right].$$

(7)

Proof. 

Let $\epsilon >0$ be given. In case $\lambda >1,$ we have

$$\left\{n\le N:\left|\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)-l\right|\ge \epsilon \right\}\subset \left\{n\le {\lambda }_N:\left|\Delta \widehat{F_n}\left(\sigma \right)-l\right|\ge \epsilon \right\}.$$

This implies that

$${\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)-l\right|\ge \epsilon \right\}\right|\le {\displaystyle \frac{\lambda }{{\lambda }_N+1}}\left|\left\{n\le {\lambda }_N:\left|\Delta \widehat{F_n}\left(\sigma \right)-l\right|\ge \epsilon \right\}\right|.$$

(8)

By taking the limits as $N\to \infty$ on both sides of the inequality (8), we obtain $St-lim\widehat{F_{{\lambda }_n}}\left(\sigma \right)=l.$

In case $0<\lambda <1.$ According to our claim, the same term $\Delta \widehat{F_j}\left(\sigma \right)$ cannot occur more than $1+{\displaystyle \frac{1}{\lambda }}$ times in the sequence $\left(\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)\right).$ It turns out that if for some integers K and $L,$ we have

$$j={\lambda }_K={\lambda }_{K+1}=\cdots ={\lambda }_{K+L-1}<{\lambda }_{K+L},$$

this implies that

$$j\le \lambda K<\lambda \left(k+1\right)<\cdots <\lambda \left(k+L-1\right)<j+1\le \lambda \left(K+L\right).$$

So that

$$j+\lambda \left(L-1\right)\le \lambda \left(K+L-1\right)<j+1$$

and $\lambda \left(L-1\right)<1,$ that is, $L<1+{\displaystyle \frac{1}{\lambda }}.$ Consequently,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)-l\right|\ge \epsilon \right\}\right|& \le \left(1+{\displaystyle \frac{1}{\lambda }}\right){\displaystyle \frac{{\lambda }_N+1}{N+1}}{\displaystyle \frac{1}{{\lambda }_N+1}}\left|\left\{n\le {\lambda }_N:\left|\Delta \widehat{F_n}\left(\sigma \right)-l\right|\ge \epsilon \right\}\right|\hfill \\ \hfill & \le {\displaystyle \frac{2\left(\lambda +1\right)}{{\lambda }_N+1}}\left|\left\{n\le {\lambda }_N:\left|\Delta \widehat{F_n}\left(\sigma \right)-l\right|\ge \epsilon \right\}\right|,\hfill \end{array}$$

(9)

provided N is large enough in the sense that ${\displaystyle \frac{{\lambda }_N+1}{N+1}}\le 2\lambda .$ By taking the limits as $N\to \infty$ on both sides of the inequality (9), we obtain that $St-lim\widehat{F_{{\lambda }_n}}\left(\sigma \right)=l.$ □

Lemma 3.

If a sequence $\left(x_k\right)$ is Δ-Fibonacci statistically summable $\left(\Delta \widehat{F},C,1\right)$ to a number $l,$ then for each $\lambda >1,$

$$St-lim{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\Delta \widehat{F_k}\left(x\right)=l;$$

(10)

and for each $0<\lambda <1,$

$$St-lim{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\Delta \widehat{F_k}\left(x\right)=l.$$

(11)

Proof. 

Let $\lambda >1.$ Then, by the definition of $\Delta \widehat{F_n}\left(\sigma \right)$ and using the fact ${\lambda }_n>n$ for $\lambda >1,$ then we have

$${\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\Delta \widehat{F_k}\left(x\right)=\Delta \widehat{F_n}\left(\sigma \right)+{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left(\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)-\Delta \widehat{F_n}\left(\sigma \right)\right).$$

(12)

Now, (10) follows from (2) and Lemmas 1 and 2, and the fact that for large enough $n,$

$${\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\le {\displaystyle \frac{2\lambda }{\lambda -1}}.$$

(13)

For $0<\lambda <1.$ We use this equality this time around:

$${\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k=n+1}^n\Delta \widehat{F_k}\left(x\right)=\Delta \widehat{F_n}\left(\sigma \right)+{\displaystyle \frac{{\lambda }_n+1}{n-{\lambda }_n}}\left(\Delta \widehat{F_n}\left(\sigma \right)-\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)\right),$$

(14)

provided $0<\lambda <1$ and n is the large enough in the sense that ${\lambda }_n<n;$ and the following inequality for $n,$

$${\displaystyle \frac{{\lambda }_n+1}{n-{\lambda }_n}}\le {\displaystyle \frac{2\lambda }{1-\lambda }}.$$

(15)

□

Now, we provide two-sided Tauberian conditions for sequences of complex numbers.

Theorem 7.

Let $\left(x_k\right)$ be a sequence of complex numbers that is Δ-Fibonacci statistically summable $\left(\Delta \widehat{F},C,1\right)$ to a finite limit. Then, $\left(x_k\right)$ is $\left(\Delta \widehat{F},S\right)$-convergent to the same limit if and only if for each $\epsilon >0,$

$$\underset{\lambda >1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\right|\ge \epsilon \right\}\right|=0$$

(16)

or

$$\underset{0<\lambda <1}{inf}\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)\right|\ge \epsilon \right\}\right|=0.$$

(17)

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 $\lambda >1$ and (6) for $0<\lambda <1.$

Conversely, assume that (2)–(4) hold. To prove that (1) holds. We will prove that

$$St-lim\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right)=0.$$

(18)

Now, for case $\lambda >1,$ it follows from (12) that

$$\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)={\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left(\Delta \widehat{F_{{\lambda }_n}}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right)-{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right).$$

(19)

So, for any $\epsilon >0,$

$$\begin{array}{cc}\hfill \left\{n\le N:\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\ge \epsilon \right\}& \subset \left\{n\le N:{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left(\Delta \widehat{F_{{\lambda }_n}}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right)\ge {\displaystyle \frac{\epsilon }{2}}\right\}\hfill \\ \hfill & \cup \left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\le -{\displaystyle \frac{\epsilon }{2}}\right\}.\hfill \end{array}$$

(20)

Choose any $\delta >0.$ By (3), there exists $\lambda >1$, such that

$$\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\le -{\displaystyle \frac{\epsilon }{2}}\right\}\right|\le \delta .$$

(21)

By using Lemmas 1 and 2 and (13), we have

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left(\Delta \widehat{F_{{\lambda }_n}}\left(\sigma \right)-\Delta \widehat{F_n}\left(\sigma \right)\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\right|=0.$$

(22)

From (20)–(22), we have

$$\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\ge \epsilon \right\}\right|\le \delta .$$

This means that

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\ge \epsilon \right\}\right|=0.$$

(23)

Next, for case $0<\lambda <1.$ By (14), we may write

$$\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)={\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left(\Delta \widehat{F_n}\left(\sigma \right)-\Delta \widehat{F_{{\lambda }_n}}\left(x\right)\right)+{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right).$$

(24)

That is,

$$\begin{array}{cc}\hfill \left\{n\le N:\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\le -\epsilon \right\}& \subset \left\{n\le N:{\displaystyle \frac{{\lambda }_n+1}{n-{\lambda }_n}}\left(\Delta \widehat{F_n}\left(\sigma \right)-\Delta \widehat{F_{{\lambda }_n}}\left(x\right)\right)\ge {\displaystyle \frac{\epsilon }{2}}\right\}\hfill \\ \hfill & \cup \left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)\le -{\displaystyle \frac{\epsilon }{2}}\right\}.\hfill \end{array}$$

Similarly, by using Lemmas 1 and 2, (4) and (15), we obtain

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\le -\epsilon \right\}\right|=0.$$

(25)

Combining (23) and (25), we obtain

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge \epsilon \right\}\right|=0.$$

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 $\lambda >1$ and (6) for all $0<\lambda <1.$

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 $\epsilon >0$ be given. In case $\lambda >1,$ by (19), we get

$$\begin{array}{cc}\hfill \left\{n\le N:\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge \epsilon \right\}& \subset \left\{n\le N:{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}\left|\Delta \widehat{F_{{\lambda }_n}}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\hfill \\ \hfill & \cup \left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\left|\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}.\hfill \end{array}$$

(26)

In case $0<\lambda <1,$ by (24), we obtain

$$\begin{array}{cc}\hfill \left\{n\le N:\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge \epsilon \right\}& \subset \left\{n\le N:{\displaystyle \frac{{\lambda }_n+1}{n-{\lambda }_n}}\left|\Delta \widehat{F_n}\left(\sigma \right)-\Delta \widehat{F_{{\lambda }_n}}\left(x\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\hfill \\ \hfill & \cup \left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k={\lambda }_n+1}^n\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}.\hfill \end{array}$$

(27)

Choose any $\delta >0.$ By (16), there exists $\lambda >1$, such that

$$\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{{\lambda }_n-n}}\left|\sum _{k=n+1}^{{\lambda }_n}\left(\Delta \widehat{F_k}\left(x\right)-\Delta \widehat{F_n}\left(x\right)\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\right|\le \delta .$$

Or, by (17), there exists $0<\lambda <1$, such that

$$\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:{\displaystyle \frac{1}{n-{\lambda }_n}}\left|\sum _{k={\lambda }_n+1}^n\left(\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_k}\left(x\right)\right)\right|\ge {\displaystyle \frac{\epsilon }{2}}\right\}\right|\le \delta .$$

From (26) and (27), as well as Lemmas 1 and 2, in either case, we obtain

$$\underset{N\to \infty }{lim}sup{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge \epsilon \right\}\right|\le \delta .$$

Therefore,

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N+1}}\left|\left\{n\le N:\left|\Delta \widehat{F_n}\left(x\right)-\Delta \widehat{F_n}\left(\sigma \right)\right|\ge \epsilon \right\}\right|=0.$$

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 $\Delta$-Fibonacci convergence, $\Delta$-Fibonacci boundedness, $\Delta$-Fibonacci statistical convergence, strong $\Delta$-Fibonacci summability, and $\Delta$-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 $\Delta \left(q\right)$-Fibonacci statistical convergence, strong $\Delta \left(q\right)$-Fibonacci summability, and $\Delta \left(q\right)$-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 $\Delta$-Fibonacci statistical convergence with respect to modulus functions and strong $\Delta$-Fibonacci summability with respect to modulus functions can be introduced as a new research paper (for more details on modulus function, see [28,29]).