Turkish Journal of Analysis and Number Theory, 2020, Vol. 8, No. 6, 107-112 Available online at http://pubs.sciepub.com/tjant/8/6/2 Published by Science and Education Publishing DOI:10.12691/tjant-8-6-2 On the Entire Paranormed Triple Sequence Spaces Defined by Binomial Poisson Matrix N. SUBRAMANIAN 1 , A. ESI 2,* , TVG. SHRIPRAKASH 1 1 School of Arts Sciences and Humanities, Department of Mathematics, SASTRA, Deemed to be University, Thanjavur-613 401, India 2 Department of Basic Engineering Sciences, Malatya Turgut Ozal University, Malatya-44040, Turkey *Corresponding author: Received December 05, 2020; Revised December 16, 2020; Accepted December 23, 2020 Abstract In this paper the entire triple sequence space are the generalization of the classical Maddox's paranormed sequence space have been introduced and investigated some topological properties of entire triple sequence space of binomial Poisson matrix of   3  and   3  . Keywords: Poisson matrix, triple sequence, paranormed space, entire space Cite This Article: N. SUBRAMANIAN, A. ESI, and TVG. SHRIPRAKASH, “On the Entire Paranormed Triple Sequence Spaces Defined by Binomial Poisson Matrix.” Turkish Journal of Analysis and Number Theory, vol. 8, no. 6 (2020): 107-112. doi: 10.12691/tjant-8-6-2. 1. Introduction A triple sequence (real or complex) can be defined as a function ( ) : , x × × →     where ,  and  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. [1,2], Esi et al. [3,4,5,6,7,8], Dutta et al. [9], Subramanian et al. [10], Debnath et al. [11] and many others. Throughout , w Γ and Λ denote the classes of all, entire and analytic scalar valued single sequences, respectively. We write 3 w for the set of all complex triple sequences ( ), mnk x where ,, , mnk ∈  the set of positive integers. Then, 3 w is a linear space under the coordinate wise addition and scalar multiplication. Let ( ) mnk x be a triple sequence of real or complex numbers. Then the series ,, 1 mnk mnk x ∞ = ∑ is called a triple series. The triple series ,, 1 mnk mnk x ∞ = ∑ is said to be convergent if and only if the triple sequence ( ) mnk S is convergent, where ,, ,, 1 ( ,, 1, 2,3,...). mnk mnk ijq ijq S x mnk = = = ∑ A sequence ( ) mnk x x = is said to be triple analytic if 1 ,, . m n k mnk mnk sup x ++ <∞ The vector space of all triple analytic sequences are usually denoted by 3 Λ . A sequence ( ) mnk x x = is called triple entire sequence if 1 0 m n k mnk x ++ → as ,, . mnk →∞ The vector space of all triple entire sequences are usually denoted by 3 . Γ The space 3 Λ and 3 Γ is a metric space with the metric 1 ,, (, ) : ,, :1,2,3,... , m n k mnk mnk mnk dxy sup x y mnk ++     = −       (1.1) for all { } mnk x x = and { } mnk y y = in 3 . Γ Consider a triple sequence ( ). mnk x x = The th mnk section [ ,,] mnk x of the sequence is defined by [ ,,] ,, ,, 0 mnk mnk ijq ijq ijq x x δ = = ∑ for all ,, , mnk ∈  0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 mnk δ         =                  mnk δ has 1 in the th mnk position, and zero otherwise. The Poisson matrix is defined by . A T I I T = ⊗ + ⊗ Example: If 4 1 0 1 4 1 0 1 4 T −     = − −     −   and 1 0 0 0 1 0 0 0 1 T     =       then