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\begin{document}
\title[Triple Almost $\left( \lambda _{m_{i}}\mu _{n_{\ell }}\gamma
_{k_{j}}\right) $ Lacunary Riesz $\chi _{R_{\lambda _{m_{i}}\mu _{n_{\ell
}}\gamma _{k_{j}}}}^{3}$ defined by Orlicz function]{Triple Almost $\left(
\lambda _{m_{i}}\mu _{n_{\ell }}\gamma _{k_{j}}\right) $ Lacunary Riesz $%
\chi _{R_{\lambda _{m_{i}}\mu _{n_{\ell }}\gamma _{k_{j}}}}^{3}$ sequence
spaces defined by Orlicz function}
\author{N. Subramanian$^{1}$ and A.Esi$^{2}$}
\address{$^1$Department of Mathematics, \\
\indent SASTRA University, \\
\indent Thanjavur-613 401, India\\
$^{2}$Ad\i yaman University, Department of Mathematics\\
02040, Adiyaman, Turkey}
\thanks{nsmaths@yahoo.com; aesi23@hotmail.com}
\keywords{ analytic sequence, Orlicz function, double sequences, Riesz
space,Riesz convergence,Pringsheim convergence. \\
\indent 2010 \textit{Mathematics Subject Classification}. 40A05,40C05,40D05.%
\\
\indent {\it Received}: \\
\indent {\it Revised}: \\
}

\begin{abstract}
In this paper we introduce a new concept for generalized almost $%
\left(\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}\right)$ convergence in $%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}}-$Riesz spaces
strong $P-$ convergent to zero with respect to an Orlicz function and
examine some properties of the resulting sequence spaces. We also introduce
and study statistical convergence of generalized almost $\left(%
\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}\right)$ convergence in $%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}}-$Riesz space and
also some inclusion theorems are discussed.
\end{abstract}

\maketitle

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\section{Introduction}

\hspace{0.5cm} Throughout $w,\chi$ and $\Lambda$ denote the classes of all,
gai and analytic scalar valued single sequences, respectively. We write $%
w^{3}$ for the set of all complex triple sequences $(x_{mnk}),$ where $%
m,n,k\in \mathbb{N},$ the set of positive integers. Then, $w^{3}$ is a
linear space under the coordinate wise addition and scalar multiplication. 
\newline
\indent We can represent triple sequences by matrix. In case of double
sequences we write in the form of a square. In the case of a triple sequence
it will be in the form of a box in three dimensional case. \newline
\indent Some initial work on double series is found in \textit{Apostol [1]}
and double sequence spaces is found in \textit{Hardy [7]}, \textit{%
Subramanian et al. [8]}, \textit{Deepmala et al. [9,10] }and many others.
Later on investigated by some initial work on triple sequence spaces is
found in \textit{sahiner et al. [11]}, \textit{Esi et al. [2-5]}, \textit{%
Savas et al. [6]} , \textit{Subramanian et al. [12]}, \textit{Prakash et al.
[13-14]} and many others. \newline
\indent Let $\left(x_{mnk}\right)$ be a triple sequence of real or complex
numbers. Then the series $\sum_{m,n,k=1}^{\infty}x_{mnk}$ is called a triple
series. The triple series $\sum_{m,n,k=1}^{\infty}x_{mnk}$ give one space is
said to be convergent if and only if the triple sequence $(S_{mnk})$is
convergent, where

\begin{center}
$S_{mnk}=\sum_{i,j,q=1}^{m,n,k}x_{ijq}(m,n,k=1,2,3,. . . )$ .
\end{center}

A sequence $x=(x_{mnk})$is said to be triple analytic if

\begin{center}
$sup_{m,n,k}\left|x_{mnk}\right|^{\frac{1}{m+n+k}}<\infty.$
\end{center}

The vector space of all triple analytic sequences are usually denoted by $%
\Lambda^{3}$. A sequence $x=(x_{mnk})$ is called triple entire sequence if

\begin{center}
$\left|x_{mnk}\right|^{\frac{1}{m+n+k}}\rightarrow 0$ as $m,n,k\rightarrow
\infty.$
\end{center}

The vector space of all triple entire sequences are usually denoted by $%
\Gamma^{3}.$ Let the set of sequences with this property be denoted by $%
\Lambda^{3}$ and $\Gamma^{3}$ is a metric space with the metric 
\begin{equation}
d(x,y)=sup_{m,n,k}\left\{\left|x_{mnk}-y_{mnk}\right|^{\frac{1}{m+n+k}%
}:m,n,k:1,2,3,. . . \right\},
\end{equation}
forall$\hspace{0.05cm}x=\left\{x_{mnk}\right\}$\hspace{0.05cm}and\hspace{%
0.05cm}$y=\left\{y_{mnk}\right\}in \hspace{0.05cm}\Gamma^{3}.$ Let $%
\phi=\left\{finite\hspace{0.1cm} sequences\right\}.$ \newline
\newline
\indent Consider a triple sequence $x=(x_{mnk}).$ The $(m,n,k)^{th}$ section 
$x^{[m,n,k]}$ of the sequence is defined by $x^{[m,n,k]}=\sum{%
_{i,j,q=0}^{m,n,k}}x_{ijq}\delta_{ijq}$ for all $m,n,k\in \mathbb{N},$

\begin{center}
$\delta_{mnk}= 
\begin{bmatrix}
0 & 0 & . . . 0 & 0 & . . . \\ 
0 & 0 & . . . 0 & 0 & . . . \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
0 & 0 & . . . 1 & 0 & . . . \\ 
0 & 0 & . . . 0 & 0 & . . . \\ 
&  &  &  & 
\end{bmatrix}%
$
\end{center}

with 1 in the $(m,n,k)^{th}$ position and zero otherwise. \newline
\newline
\indent A sequence $x=(x_{mnk})$ is called triple gai sequence if $%
\left((m+n+k)!\left|x_{mnk}\right|\right)^{\frac{1}{m+n+k}}\rightarrow 0$ as 
$m,n,k\rightarrow \infty.$ The triple gai sequences will be denoted by $%
\chi^{3}$.

\section{Definitions and Preliminaries}

A triple sequence $x=\left(x_{mnk}\right)$ has limit $0$ (denoted by $%
P-limx=0$) \newline
(i.e) $\left(\left(m+n+k\right)!\left|x_{mnk}\right|\right)^{1/m+n+k}
\rightarrow 0$ as $m,n,k \rightarrow \infty.$ We shall write more briefly as 
$P-convergent\hspace{0.2cm}to\hspace{0.2cm} 0.$

\subsection{Definition}

A modulus function was introduced by Nakano[15]. We recall that a modulus $f$
is a function from $\left[0,\infty\right)\rightarrow \left[0,\infty \right),$
such that \newline
(1) $f\left(x\right)=0$ if and only if $x=0$ \newline
(2) $f\left(x+y\right)\leq f\left(x\right)+f\left(y\right),$ for all $x\geq
0,\hspace{0.05cm}y\geq 0,$ \newline
(3) $f$ is increasing, \newline
(4) $f$ is continuous from the right at 0. Since $\left|f\left(x\right)-f%
\left(y\right)\right|\leq f\left(\left|x-y\right|\right),$ it follows from
here that $f$ is continuous on $\left[0,\infty\right).$

\subsection{Definition}

Let $\left( q_{rst}\right) ,\left( \overline{q_{rst}}\right) ,\left( 
\overline{\overline{q_{rst}}}\right) $ be sequences of positive numbers and 
\newline
$Q_{r}=%
\begin{bmatrix}
q_{11} & q_{12} & ... & q_{1s} & 0... \\ 
q_{21} & q_{22} & ... & q_{2s} & 0... \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
q_{r1} & q_{r2} & ... & q_{rs} & 0... \\ 
0 & 0 & ...0 & 0 & 0... \\ 
&  &  &  & 
\end{bmatrix}%
where$ $q_{11}+q_{12}+\ldots +q_{rs}\neq 0,$ \newline
$\overline{Q}_{s}=%
\begin{bmatrix}
\overline{q}_{11} & \overline{q}_{12} & ... & \overline{q}_{1s} & 0... \\ 
\overline{q}_{21} & \overline{q}_{22} & ... & \overline{q}_{2s} & 0... \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
\overline{q}_{r1} & \overline{q}_{r2} & ... & \overline{q}_{rs} & 0... \\ 
0 & 0 & ...0 & 0 & 0... \\ 
&  &  &  & 
\end{bmatrix}%
where$ $\overline{q}_{11}+\overline{q}_{12}+\ldots +\overline{q}_{rs}\neq 0,$
\newline
$\overline{\overline{Q}}_{t}=%
\begin{bmatrix}
\overline{\overline{q}}_{11} & \overline{\overline{q}}_{12} & ... & 
\overline{\overline{q}}_{1s} & 0... \\ 
\overline{\overline{q}}_{21} & \overline{\overline{q}}_{22} & ... & 
\overline{\overline{q}}_{2s} & 0... \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
. &  &  &  &  \\ 
\overline{\overline{q}}_{r1} & \overline{\overline{q}}_{r2} & ... & 
\overline{\overline{q}}_{rs} & 0... \\ 
0 & 0 & ...0 & 0 & 0... \\ 
&  &  &  & 
\end{bmatrix}%
where$ $\overline{q}_{11}+\overline{q}_{12}+\ldots +\overline{q}_{rs}\neq 0.$
Then the transformation is given by \newline
$T_{rst}=\frac{1}{\lambda _{i}\mu _{\ell }\gamma _{j}}\frac{1}{Q_{r}%
\overline{Q}_{s}\overline{\overline{Q}}_{t}}\sum_{m=1}^{r}\sum_{n=1}^{s}%
\sum_{k=1}^{t}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left( \left(
m+n+k\right) !\left\vert x_{mnk}\right\vert \right) ^{1/m+n+k}$ is called
the Riesz mean of triple sequence $x=\left( x_{mnk}\right) .$ If $%
P-lim_{rst}T_{rst}\left( x\right) =0,0\in \mathbb{R},$ then the sequence $%
x=\left( x_{mnk}\right) $ is said to be Riesz convergent to 0. If $x=\left(
x_{mnk}\right) $ is Riesz convergent to 0, then we write $P_{R}-limx=0.$

\subsection{Definition}

Let $\lambda=\left(\lambda_{m_{i}}\right),\mu=\left(\mu_{n_{\ell}}\right)$
and $\gamma=\left(\gamma_{k_{j}}\right)$ be three non-decreasing sequences
of positive real numbers such that each tending to $\infty$ and \newline
$\lambda_{m_{i}+1}\leq \lambda_{m_{i}}+1,\lambda_{1}=1,\hspace{0.2cm}
\mu_{n_{\ell}+1}\leq \mu_{n_{\ell}}+1,\mu_{1}=1\hspace{0.2cm}
\gamma_{k_{j}+1}\leq \gamma_{k_{j}}+1,\gamma_{1}=1.$ \newline
Let $I_{m_{i}}=\left[m_{i}-\lambda_{m_{i}}+1,m_{i}\right],I_{n_{\ell}}=\left[%
n_{\ell}-\mu_{n_{\ell}}+1,n_{\ell}\right]$ and $I_{k_{j}}=\left[%
k_{j}-\gamma_{k_{j}}+1,k_{j}\right].$ \newline
For any set $K\subseteq \mathbb{N}\times \mathbb{N}\times \mathbb{N},$ the
number \newline
$\delta_{\lambda, \mu, \gamma}\left(K\right)=lim_{m,n,k\rightarrow \infty}%
\frac{1}{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}\left|\left\{\left(i,j%
\right):i\in I_{m_{i}},j\in I_{n_{\ell}},k\in
I_{k_{j}},\left(i,\ell,j,\right)\in K\right\}\right|,$ is called the $%
\left(\lambda,\mu,\gamma\right)-$ density of the set $K$ provided the limit
exists.

\subsection{Definition}

A triple sequence $x=\left(x_{mnk}\right)$ of numbers is said to be $%
\left(\lambda,\mu,\gamma\right)-$ statistical convergent to a number $\xi$
provided that for each $\epsilon>0,$ \newline
$lim_{m,n,k\rightarrow \infty}\frac{1}{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}\frac{1}{Q_{i}\overline{Q}_{\ell}\overline{\overline{Q}}_{j}}%
\left|\left\{\left(i,\ell,j\right)\in I_{m_{i}n_{\ell}k_{j}}:q_{m}\overline{q%
}_{n}\overline{\overline{q}}_{k}\left|x_{mnk}-\xi\right|\geq \epsilon
\right\}\right|=0,$ \newline
(i.e) the set $K\left(\epsilon\right)=\frac{1}{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}\frac{1}{Q_{i}\overline{Q}_{\ell}\overline{\overline{Q}}_{j}}%
\left|\left\{\left(i,\ell,j\right)\in I_{m_{i}n_{\ell}k_{j}}: q_{m}\overline{%
q}_{n}\overline{\overline{q}}_{k}\left|x_{mnk}-\xi\right|\geq \epsilon
\right\}\right|$ has $\left(\lambda, \mu,\gamma\right)-$ density zero. In
this case the number $\xi$ is called the $\left(\lambda, \mu, \gamma\right)-$
statistical limit of the sequence $x=\left(x_{mnk}\right)$ and we write $%
St_{\left(\lambda, \mu, \gamma\right)}lim_{m,n,k\rightarrow \infty}=\xi.$

\subsection{Definition}

The triple sequence $\theta_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}%
\right)\right\}$ is called triple lacunary if there exist three increasing
sequences of integers such that

\begin{center}
$m_{0}=0,h_{i}=m_{i}-m_{r-1}\rightarrow \infty$ as $i\rightarrow \infty$ and 
\\[0pt]
$n_{0}=0, \overline{h_{\ell}}=n_{\ell}-n_{\ell-1}\rightarrow \infty$ as $%
\ell\rightarrow \infty.$ \\[0pt]
$k_{0}=0, \overline{h_{j}}=k_{j}-k_{j-1}\rightarrow \infty$ as $j\rightarrow
\infty.$
\end{center}

Let $m_{i,\ell,j}=m_{i}n_{\ell}k_{j},h_{i,\ell,j}=h_{i}\overline{h_{\ell}}%
\overline{h_{j}},$ and $\theta_{i,\ell,j}$ is determine by \newline
$I_{i,\ell,j}=\left\{\left(m,n,k\right):m_{i-1}< m< m_{i}\hspace{0.05cm}and%
\hspace{0.05cm}n_{\ell-1}< n\leq n_{\ell} \hspace{0.05cm}and\hspace{0.05cm}%
k_{j-1}< k\leq k_{j} \right\}, q_{k}=\frac{m_{k}}{m_{k-1}},\overline{q_{\ell}%
}=\frac{n_{\ell}}{n_{\ell-1}},\overline{q_{j}}=\frac{k_{j}}{k_{j-1}}.$ 
\newline
Using the notations of lacunary sequence and Riesz mean for triple
sequences. \newline
$\theta_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}$ be a
triple lacunary sequence and $q_{m}\overline{q}_{n}\overline{\overline{q}}%
_{k}$ be sequences of positive real numbers such that $Q_{m_{i}}=\sum_{m\in
\left(0,m_{i}\right]}p_{m_{i}},Q_{n_{\ell}}=\sum_{n\in \left(0,n_{\ell}%
\right]}p_{n_{\ell}},Q_{n_{j}}=\sum_{k\in \left(0,k_{j}\right]}p_{k_{j}}$
and $H_{i}=\sum_{m\in \left(0,m_{i}\right]}p_{m_{i}},\overline{H}=\sum_{n\in
\left(0,n_{\ell}\right]}p_{n_{\ell}},\overline{\overline{H}}=\sum_{k\in
\left(0,k_{j}\right]}p_{k_{j}}.$ Clearly, $H_{i}=Q_{m_{i}}-Q_{m_{i-1}},%
\overline{H}_{\ell}=Q_{n_{\ell}}-Q_{n_{\ell-1}},\overline{\overline{H}}%
_{j}=Q_{k_{j}}-Q_{k_{j-1}}.$ If the Riesz transformation of triple sequences
is RH-regular, and $H_{i}=Q_{m_{i}}-Q_{m_{i-1}}\rightarrow \infty$ as $%
i\rightarrow \infty,\overline{H}=\sum_{n\in \left(0,n_{\ell}\right]%
}p_{n_{\ell}}\rightarrow \infty$ as $\ell\rightarrow \infty,\overline{%
\overline{H}}=\sum_{k\in \left(0,k_{j}\right]}p_{k_{j}}\rightarrow \infty$
as $j\rightarrow \infty,$ then $\theta^{^{\prime
}}_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}=\left\{%
\left(Q_{m_{i}}Q_{n_{j}}Q_{k_{k}}\right)\right\}$ is a triple lacunary
sequence. If the assumptions $Q_{r}\rightarrow \infty$ as $r\rightarrow
\infty,$ $\overline{Q}_{s}\rightarrow \infty$ as $s\rightarrow \infty$ and $%
\overline{\overline{Q}}_{t}\rightarrow \infty$ as $t\rightarrow \infty$ may
be not enough to obtain the conditions $H_{i}\rightarrow \infty$ as $%
i\rightarrow \infty,\overline{H}_{\ell}\rightarrow \infty$ as $%
\ell\rightarrow \infty$ and $\overline{\overline{H}}_{j}\rightarrow \infty$
as $j\rightarrow \infty$ respectively. For any lacunary sequences $%
\left(m_{i}\right),\left(n_{\ell}\right)$ and $\left(k_{j}\right)$ are
integers. \newline
Throughout the paper, we assume that $Q_{r}=q_{11}+q_{12}+\ldots +
q_{rs}\rightarrow \infty \left(r\rightarrow \infty\right),\overline{Q}_{s}=%
\overline{q}_{11}+\overline{q}_{12}+\ldots +\overline{q}_{rs}\rightarrow
\infty \left(s\rightarrow \infty\right),\overline{\overline{Q}}_{t}=%
\overline{\overline{q}}_{11}+\overline{\overline{q}}_{12}+\ldots +\overline{%
\overline{q}}_{rs}\rightarrow \infty \left(t\rightarrow \infty\right),$ such
that $H_{i}=Q_{m_{i}}-Q_{m_{i-1}}\rightarrow \infty$ as $i\rightarrow \infty,%
\overline{H}_{\ell}=Q_{n_{\ell}}-Q_{n_{\ell-1}}\rightarrow \infty$ as $%
\ell\rightarrow \infty$ and $\overline{\overline{H}}%
_{j}=Q_{k_{j}}-Q_{k_{j-1}}\rightarrow \infty$ as $j\rightarrow \infty.$ 
\newline
Let $Q_{m_{i},n_{\ell},k_{j}}=Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{%
\overline{Q}}_{k_{j}},H_{i\ell j}=H_{i}\overline{H}_{\ell}\overline{%
\overline{H}}_{j}, \newline
I^{^{\prime }}_{i\ell j}=\left\{\left(m,n,k\right):Q_{m_{i-1}}< m< Q_{m_{i}},%
\overline{Q}_{n_{\ell-1}}< n< Q_{n_{\ell}}\hspace{0.1cm}and\hspace{0.1cm}%
\overline{Q}_{k_{j-1}}< k< \overline{Q}_{k_{j}}\right\},\newline
V_{i}=\frac{Q_{m_{i}}}{Q_{m_{i-1}}},\overline{V}_{\ell}=\frac{Q_{n_{\ell}}}{%
Q_{n_{\ell-1}}}$ and $\overline{\overline{V}}_{j}=\frac{Q_{k_{j}}}{%
Q_{k_{j-1}}}.$ and $V_{i\ell j}=V_{i}\overline{V}_{\ell}\overline{\overline{V%
}}_{j}.$ \newline
If we take $q_{m}=1,\overline{q}_{n}=1\hspace{0.1cm} and \hspace{0.1cm}%
\overline{\overline{q}}_{k}=1$ for all $m,n$ and $k$ then $H_{i\ell
j},Q_{i\ell j},V_{i\ell j}$ and $I^{^{\prime }}_{i\ell j}$ reduce to $%
h_{i\ell j},q_{i\ell j},v_{i\ell j}$ and $I_{i\ell j}.$ \newline
\indent Let $f$ be an Orlicz function and $p=\left(p_{mnk}\right)$ be any
factorable triple sequence of strictly positive real numbers, we define the
following sequence spaces: \newline
$\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i%
\ell j},q,f,p\right]= \newline
\left\{P-lim_{i,\ell, j\rightarrow \infty}\frac{1}{\lambda_{m_{i}}\mu_{n_{%
\ell}}\gamma_{k_{j}}}\frac{1}{H_{i,\ell j}}\sum_{i\in I_{i\ell
j}}\sum_{\ell\in I_{i\ell j}}\sum_{j\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell,k+j}\right|\right)^{p_{mnk}}\right]\right\}$=\newline
0, uniformly in $i,\ell$ and $j.$ \newline
$\left[\Lambda^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},%
\theta_{i\ell j},q,f,p\right]= \newline
\left\{x=\left(x_{mnk}\right):P-sup_{i,\ell, j}\frac{1}{\lambda_{m_{i}}%
\mu_{n_{\ell}}\gamma_{k_{j}}}\frac{1}{H_{i,\ell j}}\sum_{i\in I_{i\ell
j}}\sum_{\ell\in I_{i\ell j}}\sum_{j\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[f\left|x_{m+i,n+\ell,k+j}\right|^{p_{mnk}}%
\right]< \infty\right\},$ uniformly in $i,\ell$ and $j.$ \newline
Let $f$ be an Orlicz function, $p=p_{mnk}$ be any factorable double sequence
of strictly positive real numbers and and $q_{m},\overline{q}_{n}$ and $%
\overline{\overline{q}}_{k}$ be sequences of positive numbers and $%
Q_{r}=q_{11}+\cdots q_{rs}$, $\overline{Q}_{s}=\overline{q}_{11}\cdots 
\overline{q}_{rs}$ and $\overline{\overline{Q}}_{t}=\overline{\overline{q}}%
_{11}\cdots \overline{\overline{q}}_{rs},$ \newline
If we choose $q_{m}= 1,\overline{q}_{n}= 1$ and $\overline{\overline{q}}%
_{k}=1$ for all $m,n$ and $k$, then we obtain the following sequence spaces. 
\newline
$\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},q,f,p%
\right]= \newline
\left\{P-lim_{i,\ell,j \rightarrow \infty}\frac{1}{\lambda_{i}\mu_{\ell}%
\gamma_{j}}\frac{1}{Q_{i}\overline{Q}_{\ell}\overline{\overline{Q}}_{j}}%
\sum_{m=1}^{i}\sum_{n=1}^{\ell}\sum_{k=1}^{j} q_{m}\overline{q}_{n}\overline{%
\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}%
\right|\right)^{p_{mnk}}\right]=0\right\},$ uniformly in $i,\ell$ and $j.$ 
\newline
$\left[\Lambda^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},q,f,p%
\right]= \newline
\left\{P-sup_{i,\ell,j}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{%
Q_{i}\overline{Q}_{\ell}\overline{\overline{Q}}_{j}}\sum_{m=1}^{i}%
\sum_{n=1}^{\ell}\sum_{k=1}^{j} q_{m}\overline{q}_{n}\overline{\overline{q}}%
_{k}\left[f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|%
\right)^{p_{mnk}}\right]<\infty\right\},$ uniformly in $i,\ell$ and $j.$

\section{Main Results}

\subsection{Theorem}

If $f$ be any Orlicz function and a bounded factorable positive triple
number sequence $p_{mnk}$ then $\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{%
\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q,f,p\right]$ is linear space 
\newline
Proof: The proof is easy. Theorefore omit the proof.

\subsection{Theorem}

For any Orlicz function $f,$ we have \newline
$\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i%
\ell j},q,f,p\right]\subset \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q,p\right].$ \newline
Proof: Let $x\in \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q,p\right]$ so that for each $i,\ell$ and $%
j$ \newline
$\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i%
\ell j},q,f,p\right]= \newline
\left\{P-lim_{i,\ell, j\rightarrow \infty}\frac{1}{\lambda_{i}\mu_{\ell}%
\gamma_{j}}\frac{1}{H_{i,\ell j}}\sum_{i\in I_{i\ell j}}\sum_{\ell\in
I_{i\ell j}}\sum_{j\in I_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q%
}}_{k}\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|%
\right)^{p_{mnk}}\right]=0\right\},$ uniformly in $i,\ell$ and $j.$ \newline
Since $f$ is continuous at zero, for $\varepsilon>0$ and choose $\delta$
with $0<\delta <1$ such that $f\left(t\right)< \epsilon$ for every $t$ with $%
0\leq t \leq \delta.$ We obtain the following, $\newline
\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{h_{i\ell j}}\left(h_{i\ell
j}\epsilon\right)+\newline
\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{h_{i\ell j}}\sum_{m\in
I_{i,\ell,j}}\sum_{n\in I_{i,\ell,j}}\sum_{k\in I_{i,\ell, j}\hspace{0.05cm}%
and\hspace{0.05cm}\left|x_{m+i,n+\ell,k+j}-0\right|> \delta} f\left[%
\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]\newline
\frac{1}{h_{i\ell j}}\left(h_{i\ell j}\epsilon\right)+ \frac{1}{%
\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{h_{i\ell j}}K\delta^{-1}f\left(2%
\right)h_{i\ell j}\hspace{0.05cm}\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{%
\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q,p\right].$ \newline
Hence $i,\ell$ and $j$ goes to infinity, we are granted $x\in \left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell
j},q,f,p\right].$

\subsection{Theorem}

Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple
lacunary sequence and $q_{i},\overline{q}_{\ell} \overline{\overline{q}}_{j}$
with $liminf_{i}V_{i}>1,\hspace{0.2cm}liminf_{\ell}\overline{V_{\ell}}>1%
\hspace{0.2cm}and\hspace{0.2cm}liminf_{j}V_{j}>1$ then for any Orlicz
function $f,\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},,f,q,p\right]\subseteq \left[\chi^{3}_{R_{\lambda_{m_{i}}%
\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q,p\right]$ \newline
\textbf{Proof: } Suppose $liminf_{i}V_{i}>1,\hspace{0.2cm}liminf_{\ell}%
\overline{V_{\ell}}>1$ and $liminf_{j}\overline{\overline{V}}_{j}>1$ then
there exists $\delta>0$ such that $V_{i}>1+\delta,\hspace{0.2cm}\overline{%
V_{\ell}}>1+\delta$ and $\overline{\overline{V}}_{j}>1+\delta.$ This implies 
$\frac{H_{i}}{Q_{m_{i}}}\geq \frac{\delta}{1+\delta},\hspace{0.2cm}\frac{%
\overline{H}_{\ell}}{\overline{Q}_{n_{\ell}}}\geq \frac{\delta}{1+\delta}$
and $\frac{\overline{\overline{H}}_{j}}{\overline{\overline{Q}}_{k_{j}}}\geq 
\frac{\delta}{1+\delta}$ Then for $x\in \left[\chi^{3}_{R_{\lambda_{m_{i}}%
\mu_{n_{\ell}}\gamma_{k_{j}}}},f,q,p\right],$ we can write for each $i,\ell$
and $j.$ \newline
$A_{i,\ell, j}=\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{H_{i\ell j}}%
\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell,
j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k%
\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}=%
\newline
\indent\hspace{0.8cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{%
H_{i\ell j}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}-\newline
\indent\hspace{0.8cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{%
H_{i\ell j}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}%
\sum_{k=1}^{k_{i-1}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[%
f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}}-\newline
\indent\hspace{0.8cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{%
H_{i\ell j}}\sum_{m=m_{i-1}+1}^{m_{i}}\sum_{n=1}^{n_{\ell-1}}%
\sum_{k=1}^{k_{j-1}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[%
f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}}-\newline
\indent\hspace{0.8cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{%
H_{i\ell j}}\sum_{k=k_{j}+1}^{k_{j}}\sum_{n=n_{\ell-1}+1}^{n_{\ell}}%
\sum_{m=1}^{m_{k-1}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[%
f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}} \newline
\indent\hspace{0.1cm}=\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{Q}_{k_{j}}}{Hh_{i\ell j}}\newline
\left(\frac{1}{Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{\overline{Q}}%
_{k_{j}}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}} q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-%
\newline
\indent\hspace{0.7cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
Q_{m_{k-1}}\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}{%
H_{i\ell j}}\newline
\left(\frac{1}{Q_{m_{i-1}}\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k%
{j-1}}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}f\left[%
\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}}\right)\newline
-\indent\hspace{0.7cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
Q_{k_{j-1}}}{H_{i\ell j}}\left(\frac{1}{Q_{k_{j-1}}}%
\sum_{m=m_{i-1}+1}^{m_{i}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j}}f\left[%
\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}}\right)\newline
-\indent\hspace{0.7cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
\overline{Q}_{n_{\ell-1}}}{H_{i\ell j}}\left(\frac{1}{\overline{Q}%
_{n_{\ell-1}}}\sum_{m=m_{k-1}+1}^{m_{k}}\sum_{n=1}^{n_{\ell-1}}%
\sum_{k=1}^{k_{j}}f\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}%
\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-\newline
\indent\hspace{0.7cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
\overline{\overline{Q}}_{m_{k-1}}}{H_{i\ell j}}\left(\frac{1}{\overline{%
\overline{Q}}_{m_{k-1}}}\sum_{k=1}^{k_{j}}\sum_{n=n_{\ell-1}+1}^{n_{\ell}}%
\sum_{m=1}^{m_{k-1}}f\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}%
\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right).$ \newline
Since $x\in \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},f,q,p\right],$ the last three terms tend to zero uniformly
in $m,n,k$ in the sense, thus, for each $i,\ell$ and $j$ \newline
$A_{i,\ell ,j}=\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{Q_{m_{i}}%
\overline{Q}_{n_{\ell}}\overline{\overline{Q}}_{k_{j}}}{H_{i\ell j}}\newline
\left(\frac{1}{Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{\overline{Q}}%
_{k_{j}}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-%
\newline
\indent\hspace{0.7cm}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{%
Q_{m_{i-1}}\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}{%
H_{i\ell j}}\newline
\left(\frac{1}{Q_{m_{i-1}}\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}%
_{k_{j-1}}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}%
\sum_{k=1}^{k_{j-1}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[%
f\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}%
\right]^{p_{mnk}}\right)+O\left(1\right).$ \newline
Since $\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}H_{i\ell j}=\frac{1}{%
\lambda_{i}\mu_{\ell}\gamma_{j}}Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{%
\overline{Q}}_{k_{j}}-\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}$ we are granted
for each $i,\ell$ and $j$ the following

\begin{center}
$\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{Q_{m_{i}}\overline{Q}%
_{n_{\ell}}\overline{\overline{Q}}_{k_{j}}} {H_{i\ell j}}\leq \frac{1+\delta%
}{\delta}$ and $\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}} {H_{i\ell j}}%
\leq \frac{1}{\delta}.$
\end{center}

The terms \newline
$\left(\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{Q_{m_{i}}\overline{Q%
}_{n_{\ell}}\overline{\overline{Q}}_{k_{j}}} \sum_{m=1}^{m_{i}}%
\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}q_{m}\overline{q}_{n}\overline{%
\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!\left|x_{m+r,n+s,k+u}%
\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)$ and \newline
$\left(\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}}\frac{1}{Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}%
\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)$
are both gai sequences for all $r,s$ and $u.$ Thus $A_{i\ell j}$ is a gai
sequence for each $i,\ell$ and $j.$ Hence $x\in \left[\chi^{3}_{R_{%
\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q,p\right].$

\subsection{Theorem}

Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple
lacunary sequence and $q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}$
with $limsup_{i}V_{i}<\infty,\hspace{0.1cm}limsup_{\ell}\overline{V}%
_{\ell}<\infty$ and $limsup_{j}\overline{\overline{V}_{j}}< \infty$ then for
any Orlicz function $f,\hspace{0.5cm}\left[\chi^{3}_{R_{\lambda_{m_{i}}%
\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q,f,p\right]\subseteq \left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},q,f,p\right].$ 
\newline
\textbf{Proof: } Since $limsup_{i}V_{i}<\infty,\hspace{0.1cm}limsup_{\ell}%
\overline{V}_{\ell}<\infty$ and $limsup_{j}\overline{\overline{V}_{j}}<
\infty$ there exists $H>0$ such that $V_{i}<H,\hspace{0.2cm}\overline{%
V_{\ell}}<H$ and $\overline{\overline{V}}_{j}<H$ for all $i,\ell$ and $j.$
Let $x\in \left[\chi^{3}_{R_{\lambda_{i}\mu_{\ell}\gamma_{j}}},\theta_{i\ell
j},q,f,p\right]$ and $\epsilon>0.$ Then there exist $i_{0}>0,\ell_{0}>0$ and 
$j_{0}>0$ such that for every $a\geq i_{0},\hspace{0.2cm}b\geq \ell_{0}$ and 
$c\geq j_{0}$ and for all $i,\ell$ and $j.$

\begin{center}
$A^{^{\prime }}_{abc}=\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{abc}}%
\sum_{m\in I_{a,b,c}}\sum_{n\in I_{a,b,c}} \sum_{k\in I_{a,b,c}}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\rightarrow
0\hspace{0.05cm}as\hspace{0.05cm}m,n,k\rightarrow \infty.$
\end{center}

Let $G^{^{\prime }}=max\left\{A^{^{\prime }}_{a,b,c}:1\leq a\leq i_{0},%
\hspace{0.2cm}1\leq b\leq \ell_{0}\hspace{0.2cm} and \hspace{0.2cm}1\leq
c\leq j_{0}\right\}$ and $p,r$ and $t$ be such that $m_{i-1}<p\leq m_{i},%
\hspace{0.2cm}n_{\ell-1}<r\leq n_{\ell}$ and $k_{j-1}<t\leq k_{j}.$ Thus we
obtain the following: \newline
$\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{p}\overline{Q}_{r}\overline{%
\overline{Q}}_{t}}\sum_{m=1}^{p}\sum_{n=1}^{r}\sum_{k=1}^{t}q_{m}\overline{q}%
_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}} \newline
\leq \frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{Q}%
_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}\sum_{m=1}^{m_{i}}%
\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\newline
\leq \frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{Q}%
_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}\sum_{a=1}^{i}\sum_{b=1}^{%
\ell}\sum_{c=1}^{j}\newline
\left(\sum_{m\in I_{a,b,c}}\sum_{n\in I_{a,b,c}}\sum_{k\in I_{a,b,c}}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)%
\newline
= \frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{Q}%
_{n_{\ell-1}}\overline{Q}_{k_{j-1}}}\sum_{a=1}^{i_{0}}\sum_{b=1}^{\ell_{0}}%
\sum_{c=1}^{j_{0}}H_{a,b,c}A^{^{\prime }}_{a,b,c}+\newline
\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{k-1}}\overline{Q}_{n_{\ell-1}}%
\overline{\overline{Q}}_{k_{j-1}}}\sum_{\left(i_{0}<a\leq i\right)\bigcup
\left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq
j\right)}H_{a,b,c}A^{^{\prime }}_{a,b,c}\newline
\leq \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{i_{0}}}\overline{%
\overline{Q}}_{k_{i_{0}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}+\frac{1}{%
\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{Q}_{n_{\ell-1}}\overline{%
\overline{Q}}_{k_{j-1}}}\sum_{\left(i_{0}<a\leq i\right)\bigcup
\left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq
\j\right)}H_{a,b,c}A^{^{\prime }}_{a,b,c}\newline
\leq \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{\ell_{0}}}\overline{%
\overline{Q}}_{k_{j_{0}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}m_{i-1}n_{%
\ell-1}k_{j-1}}+\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{%
Q}_{n_{\ell-1}}\overline{\overline{Q}}_{j_{j-1}}}\sum_{\left(i_{0}<a\leq
i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq
j\right)}H_{a,b,c}A^{^{\prime }}_{a,b,c}\newline
\leq \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{\ell_{0}}\overline{%
\overline{Q}}_{k_{j_{0}}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}+\newline
\left(sup_{a\geq i_{0}\bigcup b\geq \ell_{0}\bigcup c\geq j_{0}}A^{^{\prime
}}_{a,b,c}\right)\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}%
\sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq
\ell\right)\bigcup \left(j_{0}<c\leq j\right)}H_{a,b,c}\newline
\leq \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{\ell_{0}}\overline{%
\overline{Q}}_{k_{j_{0}}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}+\frac{\epsilon}{%
\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}\overline{Q}_{n_{\ell-1}}\overline{%
\overline{Q}}_{k_{j-1}}} \sum_{\left(i_{0}<a\leq i\right)\bigcup
\left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq j\right)}H_{a,b,c} 
\newline
= \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{\ell_{0}}\overline{%
\overline{Q}}_{k_{j_{0}}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}+V_{i}\overline{V}%
_{\ell}\overline{\overline{V}}_{j}\epsilon \newline
\leq \frac{G^{^{\prime }}Q_{m_{i_{0}}}\overline{Q}_{n_{\ell_{0}}\overline{%
\overline{Q}}_{k_{j_{0}}}}}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{m_{i-1}}%
\overline{Q}_{n_{\ell-1}}\overline{\overline{Q}}_{k_{j-1}}}+\epsilon H^{3}.$ 
\newline
Since $Q_{m_{i-1}}\hspace{0.1cm}\overline{Q}_{n_{\ell-1}}\hspace{0.1cm}%
\overline{\overline{Q}}_{k_{j-1}}\rightarrow \infty$ as $i,\ell,j\rightarrow
\infty$ approaches infinity, it follows that

\begin{center}
$\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}Q_{p}\overline{Q}_{r}\overline{%
\overline{Q}}_{t}}\sum_{m=1}^{p}\sum_{n=1}^{q}\sum_{k=1}^{t}q_{m}\overline{q}%
_{n}\overline{\overline{q}}_{k}\left[f\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}= 0,\hspace{%
0.05cm}uniformly\hspace{0.05cm}in\hspace{0.05cm}i,\ell\hspace{0.05cm}and%
\hspace{0.05cm}j.$
\end{center}

Hence $x\in \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},q,f,p\right].$

\subsection{Corollary}

Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple
lacunary sequence and $q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}$ be
sequences of positive numbers. If $1<lim_{i\ell j}V_{i\ell j}\leq lim_{i\ell
j}supV_{i\ell j}<\infty,$ then for any Orlicz function $f,\hspace{0.5cm}%
\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i%
\ell j},q,f,p\right]= \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},q,f,p\right].$

\subsection{Definition}

Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple
lacunary sequence. The triple number sequence $x$ is said to be $S_{\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]}-P$ convergent to 0 provided that for every $\epsilon>0,$ \newline
$P-lim_{i\ell J}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}%
sup_{i\ell j}\left|\left\{\left(m,n,k\right)\in I^{^{\prime }}_{i\ell
j}:q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k%
\right)!\left|x_{mnk}\right|\right)^{1/m+n+k},\bar{0}\right]\right\}\geq
\epsilon\right|=0.$ In this case we write $S_{\left[\chi^{3}_{R_{%
\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}\right]}-P-lim
x=0.$

\subsection{Theorem}

Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple
lacunary sequence. If $I^{^{\prime }}_{i,\ell, j}\subseteq I_{i,\ell, j},$
then the inclusion $\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q\right]\subset S_{\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]}$ is strict and $\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q\right]-P-lim x=S_{\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]}-P-lim x=0.$ \newline
\textbf{Proof:} Let 
\begin{equation}
K_{Q_{i\ell j}}\left(\epsilon\right)=\left|\left\{\left(m,n,k\right)\in
I^{^{\prime }}_{i\ell j}:q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}%
\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,
k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\right\}\geq \epsilon\right|
\end{equation}
Suppose that $x\in \left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q\right].$ Then for each $i,\ell$ and $j$ 
\newline
$P-lim_{i\ell j}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}%
\sum_{m\in I_{i\ell j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j }}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]=0.$ \newline
Since \newline
$\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell
j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\geq \newline
\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell
j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]=\frac{\left|K_{Q_{i\ell
j}}\left(\epsilon\right)\right|}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}$
for all $i,\ell$ and $j,$ we get $P-lim_{i,\ell, j}\frac{\left|K_{Q_{i\ell
j}}\left(\epsilon\right)\right|}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}%
=0$ for each $i,\ell$ and $j.$ This implies that $x\in S_{\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]}.$ \newline
To show that this inclusion is strict, let $x =\left(x_{mnk}\right)$ be
defined as

\begin{center}
$\left(x_{mnk}\right)=%
\begin{bmatrix}
1 & 2 & 3 & . . . \frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left[\sqrt[4]{%
H_{i,\ell, j}}\right]^{m+n+k}-1}{\left(m+n+k\right)!} & 0 & \ldots \\ 
1 & 2 & 3 & . . . \frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left[\sqrt[4]{%
H_{i,\ell, j}}\right]^{m+n+k}-1}{\left(m+n+k\right)!} & 0 & \ldots \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
\frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left[\sqrt[4]{H_{i,\ell, j}}\right]%
^{m+n+k}-1}{\left(m+n+k\right)!} & 2 & 3 & . . . \frac{\lambda_{i}\mu_{\ell}%
\gamma_{j}\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}-1}{\left(m+n+k\right)!%
} & 0 & \ldots \\ 
. &  &  &  &  &  \\ 
\frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left[\sqrt[4]{H_{i,\ell, j}}\right]%
^{m+n+k}}{\left(m+n+k\right)!} & \frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left[%
\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}}{\left(m+n+k\right)!} & \frac{%
\lambda_{i}\mu_{\ell}\gamma_{j}\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}}{%
\left(m+n+k\right)!} & 0 & \ldots &  \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
0 & 0 & 0 & . . . 0 & 0 & \ldots \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
. &  &  &  &  &  \\ 
&  &  &  &  & 
\end{bmatrix}%
;$
\end{center}

and $q_{m}=1; \overline{q}_{n}=1; \overline{\overline{q}}_{k}=1$ for all $%
m,n $ and $k.$ Clearly, $x$ is unbounded sequence. For $\epsilon>0$ and for
all $i,\ell$ and $j$ we have \newline
$\left|\left\{\left(m,n,k\right)\in I^{^{\prime }}_{i\ell j}:q_{m}\overline{q%
}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\right\}\geq
\epsilon\right|=\newline
P-lim_{i\ell j}\left(\frac{\lambda_{i}\mu_{\ell}\gamma_{j}\left(m+n+k\right)!%
\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}\left[\sqrt[4]{H_{i,\ell, j}}%
\right]^{m+n+k}\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}}{\left[\sqrt[4]{%
H_{i,\ell, j}}\right]^{m+n+k}\left(m+n+k\right)!}\right)^{1/m+n+k}=0.$ 
\newline
Therefore $x\in S_{\left[\chi^{3}_{R_{\lambda_{{m_{i}n_{\ell}k_{j}}%
}}},\theta_{i\ell j}\right]}$ with the $P-lim=0.$ Also note that \newline
$P-lim_{i\ell j}\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}%
\sum_{m\in I_{i\ell j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j }}q_{m}%
\overline{q}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]=\newline
P-\frac{1}{2}\left(lim_{i\ell j}\left(\frac{\lambda_{i}\mu_{\ell}\gamma_{j}%
\left(m+n+k\right)!\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}\left[\sqrt[4]%
{H_{i,\ell, j}}\right]^{m+n+k}\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}}{%
\left[\sqrt[4]{H_{i,\ell, j}}\right]^{m+n+k}\left(m+n+k\right)!}%
\right)^{1/m+n+k}+1\right)=\frac{1}{2}.$ Hence $x\notin \left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j},q%
\right].$

\subsection{Theorem}

Let $I^{^{\prime }}_{i\ell j}\subseteq I_{i\ell j}.$ If the following
conditions hold, then $\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q\right]_{\mu}\subset S_{\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]}$ and $\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}%
\gamma_{k_{j}}}},\theta_{i\ell j},q\right]_{\mu}-P-lim x=S_{\left[%
\chi^{3}_{R_{\lambda_{i}\mu_{\ell}\gamma_{j}}},\theta_{i\ell j}\right]%
}-P-lim x=0.$ \newline
(1). $0<\mu<1$ and $0\leq \left[\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]<1.$ \newline
(2). $1<\mu<\infty$ and $1\leq \left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]<\infty.$ 
\newline
\textbf{Proof:} Let $x=\left(x_{mnk}\right)$ be strongly $\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{n_{j}}}},\theta_{i\ell j},q%
\right]_{\mu}-$ almost $P-$ convergent to the limit 0. Since $q_{m}\overline{%
q}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]^{\mu}\geq 
\newline
q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k%
\right)!\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]$
for (1) and (2), for all $i,\ell$ and $j,$ we have \newline
$\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell
j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]^{\mu}\geq \newline
\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell
j}}\sum_{n\in I_{i\ell j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}%
\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+%
\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\geq \frac{\epsilon
\left|K_{Q_{i\ell j}}\left(\epsilon\right)\right|}{\lambda_{i}\mu_{\ell}%
\gamma_{j}H_{i\ell j}}$ where $K_{Q_{i\ell j}}\left(\epsilon\right)$ is as
in (3.1). Taking limit $i,\ell, j\rightarrow \infty$ in both sides of the
above inequality, we conclude that $S_{\left[\chi^{3}_{R_{\lambda_{m_{i}}%
\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}\right]}-P-lim x=0.$

\subsection{Definition}

A triple sequence $x=\left(x_{mnk}\right)$ is said to be Riesz lacunary of $%
\chi$ almost $P-$ convergent 0 if $P-lim_{i,\ell, j}w_{mnk}^{i\ell
j}\left(x\right)=0,$ uniformly in $i,\ell$ and $j,$ where $w_{mnk}^{i\ell
j}\left(x\right)=w_{mnk}^{i\ell j}=\frac{1}{\lambda_{i}\mu_{\ell}%
\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell j}}\sum_{n\in I_{i\ell
j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}%
\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,
k+j}\right|\right)^{1/m+n+k},\bar{0}\right].$

\subsection{Definition}

A triple sequence $\left(x_{mnk}\right)$ is said to be Riesz lacunary $\chi$
almost statistically summable to 0 if for every $\epsilon>0$ the set \newline
$K_{\epsilon}=\left\{\left(i,\ell, j\right)\in \mathbb{N}\times \mathbb{N}%
\times \mathbb{N}:\left|w_{mnk}^{i\ell j},\bar{0}\right|\geq
\epsilon\right\} $ has triple natural density zero, (i.e) $%
\delta_{3}\left(K_{\epsilon}\right)=0.$ In this we write $\left[%
\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}%
\right]_{st_{2}}-P-lim x=0.$ That is, for every $\epsilon>0,$ \newline
$P-lim_{rst}\frac{1}{rst}\left|\left\{i\leq r,\ell \leq s, j\leq
t:\left|w_{mnk}^{i\ell j},\bar{0}\right|\geq \epsilon \right\}\right|=0,$
uniformly in $i,\ell$ and $j.$

\subsection{Theorem}

Let $I^{^{\prime }}_{i\ell j}\subseteq I_{i\ell j}.$ and $q_{m}\overline{q}%
_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\leq M$ for
all $m,n,k\in \mathbb{N}$ and for each $i,\ell$ and $j.$ Let $%
x=\left(x_{mnk}\right)$ be $S_{\left[\chi^{3}_{R_{\lambda_{m_{i}}\mu_{n_{%
\ell}}\gamma_{k_{j}}}},\theta_{i\ell j}\right]}-P-lim x=0.$ Let $K_{Q_{i\ell
j}}\left(\epsilon\right)=\newline
\left|\left\{\left(m,n,k\right)\in I^{^{\prime }}_{i\ell j}:q_{m}\overline{q}%
_{n}\overline{\overline{q}}_{k}\left[\left(\left(m+n+k\right)!%
\left|x_{m+i,n+\ell, k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\right\}\geq
\epsilon\right|.$ Then \newline
$\left|w_{mnk}^{i\ell j},\bar{0}\right|=\left|\frac{1}{\lambda_{i}\mu_{\ell}%
\gamma_{j}H_{i\ell j}}\sum_{m\in I_{i\ell j}}\sum_{n\in I_{i\ell
j}}\sum_{k\in I_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}%
\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,
k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\right|\leq \newline
\left|\frac{1}{\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}\sum_{m\in
I^{^{\prime }}_{i\ell j}}\sum_{n\in I^{^{\prime }}_{i\ell j}}\sum_{k\in
I^{^{\prime }}_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}%
\left[\left(\left(m+n+k\right)!\left|x_{m+i,n+\ell,
k+j}\right|\right)^{1/m+n+k},\bar{0}\right]\right|\leq \newline
\frac{M \left|K_{Q_{i\ell j}}\left(\epsilon\right)\right|}{%
\lambda_{i}\mu_{\ell}\gamma_{j}H_{i\ell j}}+\epsilon$ for each $i,\ell$ and $%
j,$ which implies that $P-lim_{i,\ell, j}w_{mnk}^{i\ell j}\left(x\right)=0,$
uniformly $i,\ell$ and $j.$ Hence, $St_{2}-P-lim_{i\ell j}w_{mnk}^{i\ell
j}=0 $ uniformly in $i,\ell, j.$ Hence $\left[\chi^{3}_{R_{\lambda_{i}\mu_{%
\ell}\gamma_{j}}},\theta_{i\ell j}\right]_{st_{2}}-P-lim x=0.$ \newline
To see that the converse is not true, consider the triple lacunary sequence 
\newline
$\theta_{i\ell
j}\left\{\left(2^{i-1}3^{\ell-1}4^{j-1}\right)\right\},q_{m}=1,\overline{q}%
_{n}=1,\overline{\overline{q}}_{k}=1$ for all $m,n$ and $k,$ and the triple
sequence $x=\left(x_{mnk}\right)$ defined by $x_{mnk}=\frac{%
\left(-1\right)^{m+n+k}}{\left(m+n+k\right)!}$ for all $m,n$ and $k.$ 
\newline
\newline
\textbf{Competing Interests: } The authors declare that there is not any
conflict of interests regarding the publication of this manuscript.

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\end{document}