## On the triple paranormed sequence space of binomial poisson matrix

#### Citation

Esi, A., Subramanian, N., Özdemir, M. K. (2019). On the triple paranormed sequence space of binomial Poisson matrix. International Conference of Mathematical Sciences. s. 030015(1)-030015(4).#### Abstract

A triple sequence (real or complex) can be defined as a function x : N × N × N → R (C), where N, R and C
denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of
triple sequence was introduced and investigated at the initial by Sahiner et al. [15, 16], Esi et al. [3, 4, 5, 6, 7, 8],
Dutta et al. [9],Subramanian et al. [17], Debnath et al. [10] and many others. Throughout w, χ and Λ denote the
classes of all, gai and analytic scalar valued single sequences, respectively. We write w3 for the set of all complex
triple sequences (xmnk), where m, n, k ∈ N, the set of positive integers. Then, w3 is a linear space under the coordinate
wise addition and scalar multiplication.
Let (xmnk) be a triple sequence of real or complex numbers. Then the series ∞
m,n,k=1 xmnk is called a triple series.
The triple series ∞
m,n,k=1 xmnk is said to be convergent if and only if the triple sequence (S mnk) is convergent, where
S mnk = m,n,k
i, j,q=1 xi jq (m, n, k = 1, 2, 3, ...). A sequence x = (xmnk) is said to be triple analytic if supm,n,k |xmnk| 1
m+n+k <
∞.The vector space of all triple analytic sequences are usually denoted by Λ3. A sequence x = (xmnk) is called triple
entire sequence if |xmnk| 1
m+n+k → 0 as m, n, k → ∞.The vector space of all triple entire sequences are usually denoted
by Γ3. The space Λ3 and Γ3 is a metric space with the metric d(x, y) = supm,n,k
|xmnk − ymnk| 1
m+n+k : m, n, k : 1, 2, 3, ...
,
for all x = {xmnk} and y = {ymnk} in Γ3.
A sequence x = (xmnk) is called triple gai sequence if ((m + n + k)! |xmnk|) 1
m+n+k → 0 as m, n, k → ∞. The triple gai
sequences will be denoted by χ3.
Consider a triple sequence x = (xmnk). The (m, n, k)
th section x[m,n,k] of the sequence is defined by x[m,n,k] = m,n,k
i, j,q=0 xi jqδi jq for all m, n, k ∈ N,where δmnk has 1 in the (m, n, k)
th position, and zero otherwise.
The Poisson matrix is defined by A = T ⊗ I + I ⊗ T.

#### Source

International Conference of Mathematical Sciences#### Collections

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