Exploring a Novel Approach to Deferred Nörlund Statistical Convergence

Abstract

This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.

1. Introduction

Statistical convergence, first introduced independently by Fast [1] and Steinhaus [2] in the same year, provides a broader framework for understanding convergence beyond the traditional concept, with deep connections to symmetry in mathematical structures. The exploration of statistical convergence has extended to various fields, such as measure theory [3], summability theory [4,5], approximation theory [6,7], time scale [8,9,10], Fourier analysis [11] and Banach spaces [12,13]. These utilizations often leverage the symmetry properties of underlying mathematical systems, further emphasizing the significance of statistical convergence in modern mathematical research.

Statistical convergence is closely related to the natural density of subsets of $\mathbb{N}$. Let $\mathrm{{\rm Y}}$ be a subset of $\mathbb{N}$. The natural density of $\mathrm{{\rm Y}}$, denoted by $d\left(\mathrm{{\rm Y}}\right)$, is defined as

$$d\left(\mathrm{{\rm Y}}\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\{v\in \mathrm{{\rm Y}}:v\le n\}\right|,$$

if the limit exists. In this definition, $\left|\{v\in \mathrm{{\rm Y}}:v\le n\}\right|$ refers to the number of elements of $\mathrm{{\rm Y}}$ that that do not exceed n.

A sequence, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$, of numbers is defined as statistically convergent (or, S-convergent) to the number ${\mathcal{\Im }}_0$ (see [14]) if, for every $\varrho >0$, the set $\left\{k\le n:\left|{\mathcal{\Im }}_k-{\mathcal{\Im }}_0\right|>\varrho \right\}$ has natural density zero, i.e.,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|{\mathcal{\Im }}_k-{\mathcal{\Im }}_0\right|>\varrho \right\}\right|=0.$$

In this scenario, we formulate $S-lim{\mathcal{\Im }}_k={\mathcal{\Im }}_0$, where S represents the set of all S-convergent sequences. Detailed explorations and various practical applications of this concept are provided in [15,16,17,18,19].

The difference sequence spaces ${\mathcal{\ell }}_{\infty }(\Delta )$, $c(\Delta )$, and $c_0(\Delta )$ were introduced in [20] and are defined as follows:

$$c_0(\Delta )=\left\{\mathcal{\Im }=\left({\mathcal{\Im }}_k\right):\Delta \mathcal{\Im }\in c_0\right\},$$

$$c(\Delta )=\left\{\mathcal{\Im }=\left({\mathcal{\Im }}_k\right):\Delta \mathcal{\Im }\in c\right\},$$

$${\mathcal{\ell }}_{\infty }(\Delta )=\left\{\mathcal{\Im }=\left({\mathcal{\Im }}_k\right):\Delta \mathcal{\Im }\in {\mathcal{\ell }}_{\infty }\right\},$$

where $\Delta \mathcal{\Im }=(\Delta {\mathcal{\Im }}_k)=({\mathcal{\Im }}_k-{\mathcal{\Im }}_{k+1})$. Here, $c_0$, c and ${\mathcal{\ell }}_{\infty }$ denote the sets of null, convergent and bounded sequences, respectively.

Consider two sequences $\left({\mathfrak{s}}_k\right)$ and $\left({\mathfrak{t}}_k\right)$ of non-negative integers, where ${\mathfrak{s}}_k<{\mathfrak{t}}_k$ for $k\in \mathbb{N}$ and ${lim}_{k\to \infty }{\mathfrak{t}}_k=\infty$. Additionally, let $\left({\mathfrak{p}}_k\right)$ and $\left({\mathfrak{q}}_k\right)$ be sequences of non-negative real numbers. The convolution of $\left({\mathfrak{p}}_k\right)$ and $\left({\mathfrak{q}}_k\right)$ is given by

$${\Re }_k=\sum _{n={\mathfrak{s}}_k+1}^{{\mathfrak{t}}_k}{\mathfrak{p}}_n{\mathfrak{q}}_{{\mathfrak{t}}_k-n}.$$

A sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ of numbers is defined as deferred Nörlund statistically convergent to ${\mathcal{\Im }}_0$ (see [21]) if for every $\varrho >0$,

$$\underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\Re }_k}}\left|\left\{n\le {\Re }_k:{\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|=0.$$

In [22], the authors developed the notion of lacunary statistical convergence. A lacunary sequence, denoted by $\mathrm{\Phi }=\left({\vartheta }_r\right)$, is an increasing sequence of non-negative integers where ${\vartheta }_0=0$, and the difference ${\mathfrak{h}}_r={\vartheta }_r-{\vartheta }_{r-1}\to \infty$ as $r\to \infty$. The intervals corresponding to $\mathrm{\Phi }$ are defined by ${\mathfrak{T}}_r=({\vartheta }_{r-1},{\vartheta }_r]$, and we define the ratio ${\displaystyle \frac{{\vartheta }_r}{{\vartheta }_{r-1}}}$ as ${\mathfrak{g}}_r$, with ${\mathfrak{g}}_1={\vartheta }_1$ for convenience. In the research, $\mathbb{L}$ is used to represent the set of all lacunary sequences.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$. A sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is defined as lacunary statistically convergent to ${\mathcal{\Im }}_0$ if, for every $\varrho >0$,

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\mathfrak{h}}_r}}\left|\left\{k\in {\mathfrak{T}}_r:|{\mathcal{\Im }}_k-{\mathcal{\Im }}_0|\ge \varrho \right\}\right|=0.$$

Recently, lacunary sequences have become a focal point in the study of sequence spaces due to their potential to redefine and extend existing notions of convergence. For more details, see [23,24,25,26].

In [27], the author defined the concept of a modulus function as follows. A function $\sigma :[0,\infty )\to [0,\infty )$ is called a modulus function (or simply a modulus) if it satisfies the following conditions:

• $\sigma \left(x\right)=0\iff x=0$,
• $\sigma (x+y)\le \sigma \left(x\right)+\sigma \left(y\right)$ for all $x,y\in [0,\infty )$,
• $\sigma$ is non-decreasing,
• $\sigma$ is right-continuous at 0.

Based on these properties, it can be concluded that a modulus function is continuous on the interval $[0,\infty )$. A modulus function may be either bounded or unbounded. For instance, the function $\sigma \left(x\right)=x^p$ with $p\in (0,1]$ is an unbounded modulus, while the function $\sigma \left(x\right)={\displaystyle \frac{x}{x+1}}$ is a bounded modulus. In the study, the symbols $\left[\mathbb{MF}\right]$ and $\left[\mathbb{UMF}\right]$ represent the classes of all modulus functions and unbounded modulus functions, respectively. Leveraging these modulus functions under specific conditions, we develop new sequence spaces that enhance the theoretical framework while uncovering significant interrelationships. These findings emphasize the flexibility and strength of modulus functions in advancing foundational concepts.

Let $\sigma \in \left[\mathbb{UMF}\right]$. A sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is defined as $\sigma$-statistically convergent to ${\mathcal{\Im }}_0$ (see [28]) if, for every $\varrho >0$,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left(n\right)}}\sigma \left(\left|\left\{k\le n: |{\mathcal{\Im }}_k-{\mathcal{\Im }}_0|\ge \varrho \right\}\right|\right)=0.$$

Various sequence spaces have been introduced and developed by several authors using modulus functions, as detailed in [29,30,31,32,33,34].

The motivation for this research stems from the need to develop more refined and comprehensive tools for understanding sequence behavior in mathematics. Traditional concepts of convergence and summability, while foundational, often lack the flexibility to address complex sequence structures encountered in advanced applications. By introducing the deferred Nörlund lacunary density and related concepts such as $\sigma$-deferred Nörlund lacunary statistical convergence and summability, this study bridges existing gaps in the theory of sequence spaces. These new definitions extend classical frameworks, offering fresh perspectives on the interplay between sequence convergence and summability. The novelty of this work lies in its incorporation of modulus functions and lacunary refinements to derive significant results on Nörlund statistical convergence, providing deeper insights and broader applicability. Moreover, the proposed concepts have promising implications for practical applications, particularly in numerical analysis and solving large structured linear systems, highlighting their relevance and potential for advancing both theoretical and applied research.

The paper is organized as follows. Section 2 presents the foundations and principal results, including key sequences $\left({\mathfrak{t}}_k\right)$, $\left({\mathfrak{s}}_k\right)$, $\left({\mathfrak{p}}_k\right)$, and $\left({\mathfrak{q}}_k\right)$. Section 2.1 introduces novel definitions, while Section 2.2 discusses analytical and computational results. Section 3 concludes with a summary and suggestions for future research.

2. Foundations and Principal Results

In this section, we outline and discuss the principal results presented in this paper. During the section, we must remember the sequences $\left({\mathfrak{t}}_k\right)$, $\left({\mathfrak{s}}_k\right)$, $\left({\mathfrak{p}}_k\right)$, and $\left({\mathfrak{q}}_k\right)$ that were mentioned in the previous section. To prevent repetition, we will omit writing these sequences throughout this section. Now, for a lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)$, we define the following numbers:

$${\mathfrak{p}}_{{\vartheta }_r}=\sum _{n\in {\mathfrak{T}}_r}{\mathfrak{p}}_n,{\mathfrak{q}}_{{\vartheta }_r}=\sum _{n\in {\mathfrak{T}}_r}{\mathfrak{q}}_n,{\mathfrak{L}}_{{\vartheta }_r}=\sum _{n\in \left(0,{\vartheta }_r\right]}{\mathfrak{p}}_n{\mathfrak{q}}_{{\mathfrak{t}}_r-n}\mathrm{and}{\mathfrak{L}}_{{\vartheta }_{r-1}}=\sum _{n\in \left(0,{\vartheta }_{r-1}\right]}{\mathfrak{p}}_n{\mathfrak{q}}_{{\mathfrak{t}}_r-n}.$$

The interval defined by $\mathrm{\Phi }$ is denoted as ${\mathfrak{T}}_r^{*}=\left({\mathfrak{L}}_{{\vartheta }_{r-1}},{\mathfrak{L}}_{{\vartheta }_r}\right]$. The $\mathrm{\Phi }$-convolution of the sequences $\left(\mathfrak{p}k\right)$ and $\left(\mathfrak{q}k\right)$ is represented by ${\Re }_{{\vartheta }_r}^{*}$ and is given by

$${\Re }_{{\vartheta }_r}^{*}=\sum _{n\in {\mathfrak{T}}_r^{*}}{\mathfrak{p}}_n{\mathfrak{q}}_{{\mathfrak{t}}_r-n}.$$

It is clear that ${\Re }_{{\vartheta }_r}^{*}={\mathfrak{L}}_{{\vartheta }_r}-{\mathfrak{L}}_{{\vartheta }_r-1}$.

2.1. Novel Definitions and Concepts

Definition 1.

The $\left(\Delta ,\mathrm{\Phi }\right)$-deferred Nörlund mean of a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is defined by

$$\Delta {\tau }_r\left(\mathcal{\Im }\right)={\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\sum _{n\in {\mathfrak{T}}_r^{*}}{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\Delta {\mathcal{\Im }}_n.$$

Definition 2.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$, $\sigma \in \left[\mathbb{UMF}\right]$ and $\mathrm{{\rm Y}}\subset \mathbb{N}$. Then, the number $d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)$ is defined as the σ-deferred Nörlund lacunary density of the set Y and is defined by

$$d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right),$$

provided the limit exists.

Definition 3.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{UMF}\right]$. Then, the sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is defined as $\Delta \sigma$-deferred Nörlund lacunary statistically summable (or, $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable) to ${\mathcal{\Im }}_0$ if

$$\underset{r\to \infty }{lim}\Delta {\tau }_r\left(\mathcal{\Im }\right)={\mathcal{\Im }}_0.$$

Throughout the study, the space of all $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable sequences is denoted by $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi },\tau \right]$.

Definition 4.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{UMF}\right]$. Then, a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ of numbers is defined as $\Delta \sigma$-deferred Nörlund lacunary statistically convergent (or, $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent) to the number ${\mathcal{\Im }}_0$ if for every $\varrho >0$, the set $\left\{n\in \mathbb{N}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}$ has zero σ-deferred Nörlund lacunary density, i.e.,

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)=0.$$

In this scenario, we formulate $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]-lim{\mathcal{\Im }}_k={\mathcal{\Im }}_0$. The space of all $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent sequences is represented by $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$.

In case ${\mathcal{\Im }}_0=0$, we write $S_{\Delta \left(0\right)}^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$ to denote the space of all $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-null sequences. It is clear that $S_{\Delta \left(0\right)}^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$ for any lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and every $\sigma \in \left[\mathbb{UMF}\right]$.

If we take $\sigma \left(x\right)=x$, then $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergence reduces to $S_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$-convergence.

Definition 5.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{MF}\right]$. Then, the sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ of points is defined as strongly $\Delta \sigma$-deferred Nörlund lacunary summable (or, strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable) to the number ${\mathcal{\Im }}_0$ if

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)=0.$$

In this scenario, we formulate $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]-lim{\mathcal{\Im }}_k={\mathcal{\Im }}_0$. The space of all strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable sequences will be denoted by $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$.

In Definition 5, there is no condition imposed that requires the modulus function $\sigma$ to be unbounded. The definition allows for the possibility that $\sigma$ may be bounded, which provides flexibility in its application across different contexts without this restriction.

2.2. Analytical and Computational Results

Theorem 1.

Let $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$, $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\mathrm{{\rm Y}}\subset \mathbb{N}$. If

$$\underset{x\to \infty }{lim}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0,$$

(1)

then $d_{\left({\sigma }^{*},\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$ implies $d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$ in case the limit in (1) exists.

Proof. 

Suppose that ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}=\eta >0$. For each $\varrho >0$, there is $n_0\in \mathbb{R}$ for which, when $x>n_0$, we get

$$\left(\eta -\varrho \right){\sigma }^{*}\left(x\right)<\sigma \left(x\right)<\left(\eta +\varrho \right){\sigma }^{*}\left(x\right)$$

(we can select $\varrho >0$ sufficiently small such that $\eta -\varrho >0$). This implies $\sigma \left(x\right)<2\eta {\sigma }^{*}\left(x\right)$ if $x>n_0$. Now, we may write

$${\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)\ge {\displaystyle \frac{1}{2\eta }}\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right).$$

(2)

That is, (2) implies

$${\displaystyle \frac{{\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}\ge {\displaystyle \frac{1}{2\eta }}{\displaystyle \frac{\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}{\displaystyle \frac{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}$$

if $\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|>n_0$ (so $n>n_0)$. Therefore, we obtain that $d_{\left({\sigma }^{*},\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$ implies $d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$. □

The next example shows that, in general, the converse of Theorem 1 does not hold.

Example 1.

Let the lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)$ be given and $\mathrm{{\rm Y}}=\left\{1,4,9,16,\cdots \right\}$. Let us take modulus functions $\sigma \left(x\right)=x$ and ${\sigma }^{*}\left(x\right)=log(x+1)$. Obviously, ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0$. Then, for any $\varrho >0$, we have

$${\displaystyle \frac{{\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}\le {\displaystyle \frac{{\sigma }^{*}\left(\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}={\displaystyle \frac{log\left(\sqrt{{\Re }_{{\vartheta }_r}^{*}}+1\right)}{log\left({\Re }_{{\vartheta }_r}^{*}+1\right)}}\nrightarrow 0$$

as $r\to \infty$. On the other hand,

$${\displaystyle \frac{\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\le {\displaystyle \frac{\sigma \left(\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}={\displaystyle \frac{\sqrt{{\Re }_{{\vartheta }_r}^{*}}}{{\Re }_{{\vartheta }_r}^{*}}}\to 0$$

as $r\to \infty$. Therefore, $d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$ and $d_{\left({\sigma }^{*},\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)\ne 0$.

Remark 1.

In Example 1, we demonstrated that the converse of Theorem 1 is not correct in general. However, there are numerous examples where the converse of Theorem 1 holds for certain special modulus functions. To illustrate this, we recall the set Y defined in Example 1 and consider modulus functions $\sigma \left(x\right)={\sigma }^{*}\left(x\right)=\sqrt{x}$. In this case, it is straightforward to observe that ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0$, and the condition $d_{({\sigma }^{*},\mathrm{\Phi })}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$ implies $d_{(\sigma ,\mathrm{\Phi })}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$.

Theorem 2.

Let $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$, $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\mathrm{{\rm Y}}\subset \mathbb{N}$. If

$$0<\underset{x\to \infty }{lim}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}=\eta <\infty ,$$

(3)

then $d_{\left({\sigma }^{*},\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0\iff d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$.

Proof. 

Suppose that $0<{\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}=\eta <\infty$. We can express the following equality

$${\displaystyle \frac{{\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}={\displaystyle \frac{{\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}}·{\displaystyle \frac{\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}·{\displaystyle \frac{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}.$$

(4)

By using the fact of limit in (3), we get ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{{\sigma }^{*}\left(x\right)}{\sigma \left(x\right)}}={\displaystyle \frac{1}{\eta }}$. So that (4) implies

$$\underset{r\to \infty }{lim}{\displaystyle \frac{\sigma \left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}=0\iff \underset{r\to \infty }{lim}{\displaystyle \frac{{\sigma }^{*}\left(\left|\left\{i\in {\mathfrak{T}}_r^{*}:i\in \mathrm{{\rm Y}}\right\}\right|\right)}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}=0.$$

This means that $d_{\left({\sigma }^{*},\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0\iff d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0$. □

The following result follows from Theorem 2 by taking ${\sigma }^{*}\left(x\right)=x$.

Corollary 1.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\mathrm{{\rm Y}}\subset \mathbb{N}$. Then, for any $\sigma \in \left[\mathbb{UMF}\right]$ providing ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{\sigma \left(x\right)}{x}}>0$, we have

$$d_{\left(\sigma ,\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0\iff d_{\left(\mathrm{\Phi }\right)}^{(p,q)}\left(\mathrm{{\rm Y}}\right)=0.$$

Theorem 3.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{UMF}\right]$. If a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent, then it is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable.

Proof. 

The proof is straightforward. □

Theorem 4.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$. If a sequence $\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent, then it is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent in case (1) holds.

Proof. 

Suppose that (1) holds and $\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$. Taking $\mathrm{{\rm Y}}=\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}$. Then, by Theorem 1,

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)=0$$

implies

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)=0.$$

Therefore, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$. □

To demonstrate that the converse of Theorem 4 is not generally true, we provide the following example.

Example 2.

Let the lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)$ be given and define $\left(x_k\right)$ such that

$$\Delta {\mathcal{\Im }}_k=\left\{\begin{array}{ccc}k,\hfill & k=m^2& \\ \hfill & & m\in \mathbb{N}.\\ {\displaystyle \frac{k}{k^2+1}},\hfill & k\ne m^2& \end{array}\right.$$

If we take $\left({\mathfrak{p}}_k\right)=\left({\mathfrak{q}}_k\right)=\left(1\right)$ and define the modulus functions $\sigma \left(x\right)=\sqrt{x}$ and ${\sigma }^{*}\left(x\right)=log\left(x+1\right)$, then obviously (1) holds. Now, for any $\varrho >0$,

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_k\right|\ge \varrho \right\}\right|\right)\le \underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\sqrt[3]{{\Re }_{{\vartheta }_r}^{*}}\right)=\underset{r\to \infty }{lim}{\displaystyle \frac{\sqrt[3]{{\Re }_{{\vartheta }_r}^{*}}}{{\Re }_{{\vartheta }_r}^{*}}}=0.$$

This follows that $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent to 0. On the other hand,

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_k\right|\ge \varrho \right\}\right|\right)& \ge \underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\sqrt[3]{{\Re }_{{\vartheta }_r}^{*}}-1\right)\hfill \\ \hfill & =\underset{r\to \infty }{lim}{\displaystyle \frac{log\left(\sqrt[3]{{\Re }_{{\vartheta }_r}^{*}}-1\right)}{log\left({\Re }_{{\vartheta }_r}^{*}\right)}}={\displaystyle \frac{1}{3}}.\hfill \end{array}$$

That is, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is not $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent to 0.

The following result follows from Theorem 4 when ${\sigma }^{*}\left(x\right)=x$.

Corollary 2.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{UMF}\right]$. If a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$-convergent, then it is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent in case

$$\underset{x\to \infty }{lim}{\displaystyle \frac{\sigma \left(x\right)}{x}}>0,$$

(5)

that is, $S_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$.

Theorem 5.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$. Then, a sequence $\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent if and only if it is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent in case (3) holds.

Proof. 

Using Theorem 3, the proof is established by considering the set

$$\mathrm{{\rm Y}}=\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}.$$

□

Theorem 6.

Let $\sigma \in \left[\mathbb{UMF}\right]$, and let ${\mathrm{\Phi }}_1=\left({\vartheta }_r\right),{\mathrm{\Phi }}_2=\left({\varpi }_r\right)\in \mathbb{L}$ such that ${\mathfrak{T}}_r^{*}\subset {\mathfrak{W}}_r^{*}$ for each $r\in \mathbb{N}$, where ${\mathfrak{W}}_r^{*}=\left({\mathfrak{L}}_{{\varpi }_{r-1}},{\mathfrak{L}}_{{\varpi }_r}\right]$. If ${\displaystyle \underset{r\to \infty }{lim}}{\displaystyle \frac{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}=0$ and $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]$-convergent, then it is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$-convergent, i.e., $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$.

Proof. 

Suppose that $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]$-convergent and $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]-lim\mathcal{\Im }={\mathcal{\Im }}_0$. Since $\sigma \in \left[\mathbb{UMF}\right]$ and ${\mathfrak{T}}_r^{*}\subset {\mathfrak{W}}_r^{*}$ for any $r\in \mathbb{N}$, then for every $\varrho >0$, we have

$$\begin{array}{cc}& {\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{W}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & ={\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\begin{array}{c}\left|\left\{{\mathfrak{L}}_{{\varpi }_{r-1}}<n\le {\mathfrak{L}}_{{\vartheta }_{r-1}}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\\ +\left|\left\{{\mathfrak{L}}_{{\vartheta }_r}<n\le {\mathfrak{L}}_{{\varpi }_r}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\\ +\left|\left\{{\mathfrak{L}}_{{\vartheta }_{r-1}}<n\le {\mathfrak{L}}_{{\vartheta }_r}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\end{array}\right)\hfill \\ & \le {\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left({\mathfrak{L}}_{{\vartheta }_{r-1}}-{\mathfrak{L}}_{{\varpi }_{r-1}}\right)+\left({\mathfrak{L}}_{{\varpi }_r}-{\mathfrak{L}}_{\vartheta }\right)+\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & \le {\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left({\mathfrak{L}}_{{\vartheta }_{r-1}}-{\mathfrak{L}}_{{\varpi }_{r-1}}\right)+{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left({\mathfrak{L}}_{{\varpi }_r}-{\mathfrak{L}}_{\vartheta }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & \le {\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left({\mathfrak{L}}_{{\varpi }_r}-{\mathfrak{L}}_{{\varpi }_{r-1}}\right)+{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left({\mathfrak{L}}_{{\varpi }_r}-{\mathfrak{L}}_{{\varpi }_{r-1}}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & ={\displaystyle \frac{2}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left({\mathfrak{L}}_{{\varpi }_r}-{\mathfrak{L}}_{{\varpi }_{r-1}}\right)+{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & \le {\displaystyle \frac{2\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}+{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right).\hfill \end{array}$$

Since $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]-lim\mathcal{\Im }={\mathcal{\Im }}_0$, we obtain that

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{W}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)=0.$$

Therefore, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$-convergent to ${\mathcal{\Im }}_0$ and thus $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$.

   □

Theorem 7.

Let $\sigma \in \left[\mathbb{UMF}\right]$, and let ${\mathrm{\Phi }}_1=\left({\vartheta }_r\right),{\mathrm{\Phi }}_2=\left({\varpi }_r\right)\in \mathbb{L}$ such that ${\mathfrak{T}}_r^{*}\subset {\mathfrak{W}}_r^{*}$ for each $r\in \mathbb{N}$. If ${\displaystyle \underset{r\to \infty }{lim}}{\displaystyle \frac{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}>0$ and $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$-convergent, then it is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]$-convergent, i.e., $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]$.

Proof. 

Suppose that $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]$-convergent and $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]-lim\mathcal{\Im }={\mathcal{\Im }}_0$. Given any $\varrho >0$. We know that ${\mathfrak{T}}_r^{*}\subset {\mathfrak{W}}_r^{*}$ for any $r\in \mathbb{N}$ so that

$$\left|\left\{n\in {\mathfrak{W}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\ge \left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|.$$

(6)

Since $\sigma \in \left[\mathbb{UMF}\right]$, the inequality (6) implies

$$\begin{array}{cc}& {\displaystyle \frac{1}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{W}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)\hfill \\ & \ge {\displaystyle \frac{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}{\displaystyle \frac{1}{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}}\sigma \left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right).\hfill \end{array}$$

Since $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_2\right]-lim\mathcal{\Im }={\mathcal{\Im }}_0$ and ${\displaystyle \underset{r\to \infty }{lim}}{\displaystyle \frac{\sigma \left({\Re }_{{\vartheta }_r}^{*}\right)}{\sigma \left({\Re }_{{\varpi }_r}^{*}\right)}}>0$, we obtain that $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,{\mathrm{\Phi }}_1\right]-lim\mathcal{\Im }={\mathcal{\Im }}_0$. □

Theorem 8.

Let $\sigma ,{\sigma }^{*}\in \left[\mathbb{MF}\right]$ and $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$. If

$$\underset{x\in \left(0,\left.\infty \right)\right.}{sup}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}<\infty$$

(7)

and a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable, then it is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable.

Proof. 

Let us take $y^{*}={\displaystyle \underset{x\in \left(0,\left.\infty \right)\right.}{sup}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}<\infty$. Then, $0<{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}\le y^{*}$ and so $\sigma \left(x\right)\le y^{*}{\sigma }^{*}\left(x\right)$ for each $x\in \left[0,\infty \right)$. It can be observed that $y^{*}>0$. If a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$, then

$${\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\le {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{y}^{*}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)$$

for each $r\in \mathbb{N}$. This follows that $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$. □

An example is provided below to show that, in general, the converse of the mentioned theorem does not hold true.

Example 3.

Let the lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)$ be given and define $\left({\mathcal{\Im }}_k\right)$ as

$$\Delta {\mathcal{\Im }}_k=\left\{\begin{array}{cc}\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right],\hfill & atthefirst\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]integersin{\mathfrak{T}}_r^{*},\\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

where $\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]$ denotes an integral part of the real number $\sqrt{{\Re }_{{\vartheta }_r}^{*}}$. Let $\left({\mathfrak{p}}_k\right)=\left({\mathfrak{q}}_k\right)=\left(1\right)$, and take the modulus functions $\sigma \left(x\right)={\displaystyle \frac{x}{x+1}}$ and ${\sigma }^{*}\left(x\right)=x$, then ${\displaystyle \underset{x\in \left(0,\left.\infty \right)\right.}{sup}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}=1<\infty$. Now, we may write

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n\right|\right)\right)& ={\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\sigma \left(\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\right)\hfill \\ \hfill & ={\displaystyle \frac{\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]}{{\Re }_{{\vartheta }_r}^{*}\left(\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]+1\right)}}\to 0\hfill \end{array}$$

as $r\to \infty$. So, $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to 0. Whereas

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n\right|\right)\right)& ={\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]{\sigma }^{*}\left(\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\right)\hfill \\ \hfill & ={\displaystyle \frac{\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]}{{\Re }_{{\vartheta }_r}^{*}}}\to 1\hfill \end{array}$$

as $r\to \infty$. So that $\left({\mathcal{\Im }}_k\right)$ is not strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to 0.

Remark 2.

In Example 3, we provided that strong $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summability does not imply strong $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summability, in general. However, for some special modulus functions this inclusion holds. Indeed, recall the sequence $\left({\mathcal{\Im }}_k\right)$ defined in Example 3 and take the modulus functions $\sigma \left(x\right)={\sigma }^{*}\left(x\right)={\displaystyle \frac{x}{x+1}}$. Then, ${\displaystyle \underset{x\in \left(0,\left.\infty \right)\right.}{sup}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}<\infty$ and $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]-lim{\mathcal{\Im }}_k=N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]-lim{\mathcal{\Im }}_k=0$.

The following result follows from Theorem 8 when ${\sigma }^{*}\left(x\right)=x$.

Corollary 3.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma \in \left[\mathbb{MF}\right]$. If a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$-summable, then it is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable in case

$$\underset{x\in \left(0,\left.\infty \right)\right.}{sup}{\displaystyle \frac{\sigma \left(x\right)}{x}}<\infty .$$

(8)

Theorem 9.

Let $\sigma ,{\sigma }^{*}\in \left[\mathbb{MF}\right]$ and $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$. If a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable, then it is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable in case

$$\underset{x\in \left(0,\left.\infty \right)\right.}{inf}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0.$$

(9)

Proof. 

Let $z^{*}={\displaystyle \underset{x\in \left(0,\left.\infty \right)\right.}{inf}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0$. So, ${\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}\ge z^{*}$ and $z^{*}{\sigma }^{*}\left(x\right)\le \sigma \left(x\right)$ for every $x\in \left[0,\infty \right)$. If a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$, then

$${\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\le {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{\displaystyle \frac{1}{z^{*}}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)$$

for any $r\in \mathbb{N}$. Therefore, we obtain that $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$. □

The subsequent result is derived from Theorems 8 and 9.

Corollary 4.

Let $\sigma ,{\sigma }^{*}\in \left[\mathbb{MF}\right]$ and $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$. If (7) and (9) hold, then a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable if and only if it is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable.

Theorem 10.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$. If a sequence $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable, then it is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent in case (9) holds and ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{{\sigma }^{*}\left(x\right)}{x}}>0$.

Proof. 

Suppose that ${\eta }^{*}={\displaystyle \underset{x\in \left(0,\left.\infty \right)\right.}{inf}}{\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}>0$. Then, ${\displaystyle \frac{\sigma \left(x\right)}{{\sigma }^{*}\left(x\right)}}\ge {\eta }^{*}$ and so ${\eta }^{*}{\sigma }^{*}\left(x\right)\le \sigma \left(x\right)$ for every $x>0$. Given any $\varrho >0$. If $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$, then

$${\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)=0.$$

Now, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{k\in {\mathfrak{T}}_r}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\hfill \\ \hfill & \ge {\eta }^{*}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{k\in {\mathfrak{T}}_r^{*}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\hfill \\ \hfill & ={\eta }^{*}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{\begin{array}{c}k\in {\mathfrak{T}}_r^{*}\\ {\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \end{array}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\hfill \\ \hfill & +{\eta }^{*}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{\begin{array}{c}k\in {\mathfrak{T}}_r^{*}\\ {\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|<\varrho \end{array}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\hfill \\ \hfill & \ge {\eta }^{*}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{\begin{array}{c}k\in {\mathfrak{T}}_r^{*}\\ {\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \end{array}}{\sigma }^{*}\left({\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\right)\hfill \\ \hfill & \ge {\eta }^{*}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left|\left\{k\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|{\sigma }^{*}\left(\varrho \right)\hfill \\ \hfill & \ge {\displaystyle \frac{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}{{\Re }_{{\vartheta }_r}^{*}}}{\displaystyle \frac{{\sigma }^{*}\left(\varrho \right)}{{\sigma }^{*}\left(1\right)}}{\eta }^{*}{\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\left|\left\{k\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_k-n}{\mathfrak{q}}_n\left|{\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right).\hfill \end{array}$$

Since $\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$ and ${\displaystyle \underset{x\to \infty }{lim}}{\displaystyle \frac{{\sigma }^{*}\left(x\right)}{x}}>0$, we obtain that

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|\right)=0.$$

Therefore, $\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$. □

Remark 3.

The converse of Theorem 10 is not valid in general. To substantiate this claim, we provide the following example.

Example 4.

Let the lacunary sequence $\mathrm{\Phi }=\left({\vartheta }_r\right)$ be given. Recall the sequence $\left({\mathcal{\Im }}_k\right)$ defined in Example 3. Taking $\sigma \left(x\right)={\sigma }^{*}\left(x\right)=x$ and $\left({\mathfrak{p}}_k\right)=\left({\mathfrak{q}}_k\right)=\left(1\right)$. Then, for any $\varrho >0$,

$${\displaystyle \frac{1}{{\sigma }^{*}\left({\Re }_{{\vartheta }_r}^{*}\right)}}{\sigma }^{*}\left(\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n\right|\ge \varrho \right\}\right|\right)={\displaystyle \frac{\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]}{{\Re }_{{\vartheta }_r}^{*}}}\to 0$$

as $r\to \infty$. So, $\left(x_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-convergent to 0. However,

$${\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}\sigma \left({\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n\right|\right)\right)={\displaystyle \frac{\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]\left[\sqrt{{\Re }_{{\vartheta }_r}^{*}}\right]}{{\Re }_{{\vartheta }_r}^{*}}}\to 1$$

as $r\to \infty$. That is, $\left(\Delta {\mathcal{\Im }}_k\right)$ is not strongly $N_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-summable to 0.

Theorem 11.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$. If a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$, then $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$ in case (8) holds and there exists $Y\in {\mathbb{R}}^{+}$ such that ${\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\le Y$ for all $n,r\in \mathbb{N}$.

Proof. 

Suppose that $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$ and ${\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\le T$ for all $n,r\in \mathbb{N}$. Since $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]\subset S_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$ for any unbounded modulus by Corollary 2, then $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$. Now, for any $\varrho >0$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)& \le {\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{\begin{array}{c}n\in {\mathfrak{T}}_r^{*}\\ {\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \end{array}}{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{\begin{array}{c}n\in {\mathfrak{T}}_r^{*}\\ {\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|<\varrho \end{array}}{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)\hfill \\ \hfill & \le Y{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left|\left\{n\in {\mathfrak{T}}_r^{*}:{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\ge \varrho \right\}\right|+\varrho .\hfill \end{array}$$

This implies that

$$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{{\Re }_{{\vartheta }_r}^{*}}}\left(\sum _{n\in {\mathfrak{T}}_r^{*}}{\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\right)=0.$$

Therefore, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$. On the other hand, since (8) holds so that $N_{\Delta }^{\left(p,q\right)}\left[\mathrm{\Phi }\right]\subset N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$ by Corollary 3. Therefore, $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$. This completes the proof. □

From Theorems 10 and 11, we obtain the following result.

Corollary 5.

Let $\mathrm{\Phi }=\left({\vartheta }_r\right)\in \mathbb{L}$ and $\sigma ,{\sigma }^{*}\in \left[\mathbb{UMF}\right]$. Then, a sequence $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is $S_{\Delta }^{\left(p,q\right)}\left[\sigma ,\mathrm{\Phi }\right]$-convergent to ${\mathcal{\Im }}_0$ if and only if $\mathcal{\Im }=\left({\mathcal{\Im }}_k\right)$ is strongly $N_{\Delta }^{\left(p,q\right)}\left[{\sigma }^{*},\mathrm{\Phi }\right]$-summable to ${\mathcal{\Im }}_0$ in case (8) holds and there exists $T\in {\mathbb{R}}^{+}$ such that ${\mathfrak{p}}_{{\mathfrak{t}}_r-n}{\mathfrak{q}}_n\left|\Delta {\mathcal{\Im }}_n-{\mathcal{\Im }}_0\right|\le T$ for all $n,r\in \mathbb{N}$.

3. Conclusions and Suggestions for Further Studies

This study introduces several novel concepts in sequence theory, including the $\sigma$-deferred Nörlund lacunary density, $\sigma$-deferred Nörlund lacunary statistical convergence, and $S_{\Delta }^{(p,q)}[\sigma ,\mathrm{\Phi }]$-summability and convergence. Additionally, the strong $N_{\Delta }^{(p,q)}[\sigma ,\mathrm{\Phi }]$-summability is proposed, offering a more robust framework for analyzing summability and convergence in numerical sequences. These advancements extend the existing theoretical landscape, providing a deeper understanding of sequence behavior and uncovering new relationships between statistical convergence and summability. The interplay between modulus functions and lacunary refinements further enriches the study, yielding significant results and establishing connections that broaden the scope of classical theories.

Future research could focus on extending these concepts to new dimensions. For example, $S_{\Delta }^{(p,q)}[\sigma ,\mathrm{\Phi }]$-convergence and strong $N_{\Delta }^{(p,q)}[\sigma ,\mathrm{\Phi }]$-summability of order $\alpha$ (for $\alpha \in (0,1]$) represent promising directions for further exploration. These extensions could deepen the theoretical framework and provide additional tools for analyzing sequence behavior. Moreover, integrating these concepts into Korovkin-type approximation theory may lead to significant advancements in practical applications, particularly in numerical analysis and the efficient resolution of large structured linear systems. The potential for these innovations to bridge theoretical insights with real-world challenges underscores the importance of continued investigation in this field.