## Abstract

In this paper, we extend the notion presented by Braha (2020) in a higher dimension, we introduce the notion of Np,qn,m,gCn,m,g1,1,1-statistically convergence and show necessity and sufficiency conditions under which the existence of the limit st-limn,m,g→∞xn,m,g=L follows from that st-limn,m,g→∞Np,qn,m,gCn,m,g1,1,1=L. These conditions are one-sided or two-sided if xn,m,g is a sequence of real or complex numbers, respectively.

### Keywords

- Nörlund-Cesro summability method
- one-sided and two-sided Tauberian conditions
- triple statistical convergence

## 1. Introduction

The concept of statistical convergence was introduced by Fast [1] and Steinhaus [2]. Besides, in this connection, Fridy [3] showed some relation to a Tauberian condition for the statistical convergence of

Let

and

for

In this paper, we show necessary and sufficient conditions under which the existence of the limit

Given two non-negative sequences

with

We say that the sequence

Throughout this paper, we will assume that the sequences

where

implies (2), then the method

Let us consider that

and as we know,

Next, we will find some conditions under which the converse implication holds, for defined convergence. Exactly, we will prove under which conditions statistical convergence of sequences

A sequence

And we say that the sequence

Theorem 1.1 Let

** Proof:** The first part of the proof is obvious. To prove the second part, we will show the following example:

Let us define

and

On the other hand, for

From last relation follows that

Theorem 1.2 Let

** Proof:** From the fact that

We will denote

for some constant

Converse of Theorem 1.2 is not true as can be seen in the following example.

Consider that

Under this conditions, after some basic calculations we get that

## 2. Tauberian theorems under N p , q n , m , g C n , m , g 1,1,1 -statistically convergence

In this section, we show the results that we obtained. Throughout this paper,

Consider that

and in case where

The condition given by relation (13) is equivalent to the condition

** Proof:** Suppose that relation (13) is valid,

From above relation and definition of sequences

Conversely, suppose that (9) is valid. Now, let

provided

Consider that (13) is satisfied and let

and

** Proof:** We begin proving the case (10), i.e. when

From (12), definition of the sequence

Prove of (11) is made similarly to the prove of (10).

In the following theorem, we characterize the converse implication when the statistically convergence follows from its

Theorem 1.3 Let

where

and

** Proof: Necessity**: Suppose that

for every

** Sufficiency**: Consider that (14) and ((15) are satisfied. In what follows, we will prove that

where

for any

On the other hand, if

where

for any

Since

In the following theorem, we will consider the case where

Theorem 1.4 Let (13) be satisfied and let

and

** Proof: Necessity**: If both (2) and (6) hold, then Proposition 2 yields (20) for every

** Sufficiency**: Suppose that (2), (13) and one of the conditions (20) and (21) are satisfied. For any given

where

For a given

where

Since

## 3. Conclusion

In this paper, we have defined and proved new Tauberian theorems under triple statistically Nörlund-Cesáro summability, as a consequence of results showed in 2, some theorems, lemmas and corollaries can be defined and proved similarly by using _{,} and

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