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    On the triple paranormed sequence space of binomial poisson matrix
    (Maltepe Üniversitesi, 2019) Subramanian, N.; Esi, A.; Özdemir, M. K.
    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, ? and ? 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 ?3. 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 ?3. The space ?3 and ?3 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 ?3. 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 ?3. 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 jq?i jq for all m, n, k ? N,where ?mnk has 1 in the (m, n, k) th position, and zero otherwise. The Poisson matrix is defined by A = T ? I + I ? T.