Lacunary Riesz χ 3 R λmi. Key Words: Analytic sequence, Orlicz function, Double sequences, Riesz space, Riesz convergence, Pringsheim convergence.

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1 Bol Soc Paran Mat (3s) v 37 2 (209): c SPM ISSN on line ISSN in press SPM: wwwspmuembr/bspm doi:05269/bspmv37i23430 ) Triple Almost Lacunary Riesz Sequence Spaces µ nl Defined by Orlicz Function Nagarajan Subramanian and Ayhan Esi abstract: In this paper we introduce a new concept for generalized almost ) convergence in µ nl 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 convergence in χ 3 R Riesz space and also some λmi µ nl inclusion theorems are discussed Key Words: Analytic sequence, Orlicz function, Double sequences, Riesz space, Riesz convergence, Pringsheim convergence Contents Introduction 29 2 Definitions and Preliminaries 30 3 Main Results 34 Introduction Throughout w,χ and Λ 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 N, the set of positive integers Then, w 3 is a linear space under the coordinate wise addition and scalar multiplication 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 Some initial work on double series is found in Apostol and double sequence spaces is found in Hardy 7, Subramanian et al 8, Deepmala et al 9,9 and many others Later on investigated by some initial work on triple sequence spaces is found in sahiner et al, Esi et al 2,3,4,5, Savas et al 6, Subramanian et al 2, Prakash et al 3,4 and many others Let (x mnk ) be a triple sequence of real or complex numbers Then the series m,n,k= x mnk is called a triple series The triple series m,n,k= x mnk give one 200 Mathematics Subject Classification: 40A05, 40C05, 40D05 Submitted November 2, 206 Published March 5, Typeset by B S P M style c Soc Paran de Mat

2 30 N Subramanian and A Esi spaceissaidtobeconvergentifandonlyifthe triplesequence(s mnk ) isconvergent, where S mnk = m,n,k i,j,q= x ijq (m,n,k =,2,3,) A sequence x = (x mnk ) is said to be triple analytic if sup m,n,k m+n+k x mnk < The vector space of all triple analytic sequences are usually denoted by Λ 3 A sequence x = (x mnk ) is called triple entire sequence if m+n+k x mnk 0 as m,n,k The vector space of all triple entire sequences are usually denoted by Γ 3 Let the set of sequences with this property be denoted by Λ 3 and Γ 3 is a metric space with the metric d(x,y) = sup m,n,k } m+n+k x mnk y mnk : m,n,k :,2,3,, () for all x = x mnk } and y = y mnk } in Γ 3 Let φ = finitesequences} 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 = m,n,k i,j,q=0 x ijqδ ijq for all m,n,k N, δ mnk = with in the (m,n,k) th position and zero otherwise A sequence x = (x mnk ) is called triple gai sequence if m+n+k ((m+n+k)! x mnk ) 0, as m,n,k The triple gai sequences will be denoted by χ 3 2 Definitions and Preliminaries A triple sequence x = (x mnk ) has limit 0 (denoted by P limx = 0) (ie) ((m+n+k)! x mnk ) /m+n+k 0 as m,n,k We shall write more briefly as P convergent to 0

3 ) Triple Almost Lacunary Riesz definedby Orlicz Func3 µ nl Definition 2 A function M : 0, ) 0, ) is said to be an Orlicz function if it satisfies the following conditions (i) M is continuous, convex and non-decreasing; (ii) M(0) = 0,M(x) > 0 and M(x) as x Remark 22 If the convexity of an Orlicz function is replaced by M(x + y) M(x)+M(y), then this function is called modulus function Definition 23 Let (q rst ),(q rst ), ( ) q rst be sequences of positive numbers and q q 2 q s 0 q 2 q 22 q 2s 0 Q r = q r q r2 q rs where q +q 2 ++q rs 0, q q 2 q s 0 q 2 q 22 q 2s 0 Q s = q r q r2 q rs where q +q 2 ++q rs 0, q q 2 q s 0 q 2 q 22 q 2s 0 Q t = q r q r2 q rs where q +q 2 ++q rs 0 Then the transformation is given by T rst = Q r Q s Q t r s m=n=k= t ((m+n+k)! x mnk ) /m+n+k is called the Riesz mean of triple sequence x = (x mnk ) If P lim rst T rst (x) = 0, 0 R, then the sequence x = (x mnk ) is said to be Riesz convergent to 0 If x = (x mnk ) is Riesz convergent to 0, then we write P R limx = 0

4 32 N Subramanian and A Esi Definition 24 Let λ = (λ mi ),µ = ( ) ( ) µ nl and γ = be three non-decreasing sequences of positive real numbers such that each tending to and λ mi+ λ mi +,λ =, µ nl + µ nl +,µ = + +,γ = Let I mi = m i λ mi +,m i,i nl = n l µ nl +,n l and I = k j +,k j For any set K N N N, the number δ λ,µ,γ (K) = lim (i,j) : i Imi,j I nl,k I,(i,l,j,) K }, m,n,k λ mi µ nl is called the (λ,µ,γ) density of the set K provided the limit exists Definition 25 A triple sequence x = (x mnk ) of numbers is said to be (λ,µ,γ) statistical convergent to a number ξ provided that for each ǫ > 0, lim m,n,k (ie) the set K(ǫ) = λ mi µ nl Q i Q l Q j (i,l,j) Imin l k j : x mnk ξ ǫ } = 0, λ mi µ nl Q i Q l Q j (i,l,j) Imin l k j : x mnk ξ ǫ } has (λ,µ,γ) density zero In this case the number ξ is called the (λ,µ,γ) statistical limit of the sequence x = (x mnk ) and we write St (λ,µ,γ) lim m,n,k = ξ Definition 26 The triple sequence θ i,l,j = (m i,n l,k j )} is called triple lacunary if there exist three increasing sequences of integers such that and m 0 = 0,h i = m i m r as i, n 0 = 0,h l = n l n l as l k 0 = 0,h j = k j k j as j Let m i,l,j = m i n l k j,h i,l,j = h i h l h j, and θ i,l,j is determine by I i,l,j = (m,n,k) : m i < m < m i and n l < n n l and k j < k k j }, q k = m k m k,q l = n l n l,q j = k j k j Using the notations of lacunary sequence and Riesz mean for triple sequences θ i,l,j = (m i,n l,k j )} be a triple lacunary sequence and be sequences of positive real numbers such that Q mi = p mi,q nl = p nl,q nj = m (0,m i n (0,n l k (0,k j p

5 ) Triple Almost Lacunary Riesz definedby Orlicz Func33 µ nl and H i = p mi,h = p nl,h = p m (0,m i n (0,n l k (0,k j Clearly, H i = Q mi Q mi,h l = Q nl Q nl,h j = Q Q If the Riesz transformation of triple sequences is RH-regular, and H i = Q mi Q mi as i,h = n (0,n l p n l as l,h = k (0,k j p k j as j, then θ i,l,j = (m i,n l,k j )} = ( Q mi Q nj Q kk )} is a triple lacunary sequence If the assumptions Q r as r, Q s as s and Q t as t may be not enough to obtain the conditions H i as i,h l as l and H j as j respectively For any lacunary sequences (m i ),(n l ) and (k j ) are integers Throughout the paper, we assume that Q r = q +q 2 ++q rs (r ), Q s = q +q 2 ++q rs (s ), Q t = q +q 2 ++q rs (t ), such that H i = Q mi Q mi as i,h l = Q nl Q nl as l and H j = Q Q as j Let Q mi,n l,k j = Q mi Q nl Q, = H i H l H j, I ilj = (m,n,k) : Q mi < m < Q mi,q nl < n < Q nl and Q < k < Q }, V i = Qm i Q mi,v l = Qn l Q nl and V j = Q k j Q and V ilj = V i V l V j If we take q m =,q n = and q k = for all m,n and k then,q ilj,v ilj and I ilj reduce to h ilj,q ilj,v ilj and I ilj Let f be an Orlicz function and p = (p mnk ) be any factorable triple sequence of strictly positive real numbers, we define the following sequence spaces: χ 3 R λ m µ i nl γ ilj,q,f,p = i I ilj l I ilj P lim i,l,j λ mi µ nl H i,lj f ((m+n+k)! x m+i,n+l,k+j ) pmnk = 0, j I ilj

6 34 N Subramanian and A Esi uniformly in i,l and j Λ 3 R λmi µ nl γ ilj,q,f,p = x = (x mnk ) : P sup i,l,j i I ilj l I ilj λ mi µ nl H i,lj f x m+i,n+l,k+j pmnk <, j I ilj uniformly in i,l and j Let f be an Orlicz function, p = p mnk be any factorable double sequence of strictly positive real numbers and q m,q n and q k be sequences of positive numbers and Q r = q + q rs, Q s = q q rs and Q t = q q rs If we choose q m =,q n = and q k = for all m,n and k, then we obtain the following sequence spaces,q,f,p = µ nl i l m=n=k= uniformly in i,l and j m=n=k= P lim m i,l,j Q i Q l Q j } j f ((m+n+k)! x m+i,n+l,k+j ) p mnk = 0, Λ 3 R λmi,q,f,p = µ nl P sup i,l,j Q i Q l Q j } i l j f ((m+n+k)! x m+i,n+l,k+j ) p mnk <, uniformly in i,l and j 3 Main Results Theorem 3 If f be any Orlicz function and a bounded factorable positive triple number sequence p mnk then µ nl γ ilj,q,f,p is linear space Proof: The proof is easy Therefore omit the proof

7 ) Triple Almost Lacunary Riesz definedby Orlicz Func35 µ nl Theorem 32 For any Orlicz function f, we have µ nl γ ilj,q,f,p χ 3 R λmi µ nl γ ilj,q,p Proof: Let x µ nl γ ilj,q,p so that for each i,l and j µ nl γ ilj,q,f,p = P lim i,l,j H i,lj i I ilj l I ilj ((m+n+k)! x m+i,n+l,k+j ) pmnk = 0, j I ilj uniformly in i,l and j Since f is continuous at zero, for ε > 0 and choose δ with 0 < δ < such that f (t) < ǫ for every t with 0 t δ We obtain the following, (h ilj ǫ)+ h ilj h ilj m I i,l,j n I i,l,j k I i,l,j and x m+i,n+l,k+j 0 >δ h ilj (h ilj ǫ)+ h ilj Kδ f (2)h ilj f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k χ 3 R λ m i µ nl ilj,q,p Hence i,l and j goes to infinity, we are granted x µ nl γ ilj,q,f,p Theorem 33 Let θ i,l,j = m i,n l,k j } be a triple lacunary sequence and q i,q l q j with liminf i V i >, liminf l V l > and liminf j V j > then for any Orlicz function f,,f,q,p χ 3 µ nl R λmi µ nl γ ilj,q,p Proof: Suppose liminf i V i >, liminf l V l > and liminf j V j > then there exists δ > 0 such that V i > + δ, V l > + δ and V j > + δ This implies H i Q mi δ +δ, H l δ Hj Q nl +δ and δ Q +δ χ Then for x 3 R λ,f,q,p, we m µ i nl can write for each i,l and j A i,l,j = m I i,l,j n I i,l,j k I i,l,j f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk

8 36 N Subramanian and A Esi = m i m i n l k j m=n=k= n l m= n= k= m i f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk k i n l k j m=m i +n= k= k j k=k j+n=n l +m= f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk n l m k f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk = Q mi Q nl Q Hh ilj Q mi Q nl Q m i n l k j m=n=k= f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk Q mi Q nl Q Q Q nl Q mk Since x m i n l m= n= k= m i Q mk Q n l Q k j k j n l Q m=m i + n= k= m k n l k j Q nl m=m k +n=k= k j Q mk k= n=n l + m= f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk k j f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk n l f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk m k,f,q,p, the last three terms tend to zero uniformly in µ nl

9 ) Triple Almost Lacunary Riesz definedby Orlicz Func37 µ nl m,n,k in the sense, thus, for each i,l and j Q mi Q nl Q A i,l,j = m i n l k j f((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk Q mi Q nl Q m=n= k= Q mi Q Q nl k j m i n l k j f((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk Q mi Q nl Q m= n= +O() k= Since λ iµ l γ j = λ iµ l γ j Q mi Q nl Q λ iµ l γ j Q mi Q nl Q we are granted for each i,l and j the following Q mi Q nl Q +δ δ The terms Q mi Q nl Q and m i n l m=n=k= Q mi Q nl Q and Q mi Q nl Q k j δ k j f((m+n+k)! x m+r,n+s,k+u ) /m+n+k p mnk m i n l k j m= n= k= f((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk ) are both gai sequences for all r,s and u Thus A ilj is a gai sequence for each i,l and j Hence x µ nl γ ilj,q,p Theorem 34 Let θ i,l,j = m i,n l,k j } be a triple lacunary sequence and with limsup i V i <, limsup l V l < and limsup j V j < then for any Orlicz function f, µ nl γ ilj,q,f,p χ 3 R λmi µ nl,q,f,p

10 38 N Subramanian and A Esi Proof: Since limsup i V i <, limsup l V l < and limsup j V j < there exists H > 0 such that V i < H, V l < H and V j < H for all i,l and j Let x χ 3 R λi µ l γ ilj,q,f,p and ǫ > 0 Then there exist i 0 > 0,l 0 > 0 and j 0 > 0 such j that for every a i 0, b l 0 and c j 0 and for all i,l and j A abc = q m q H n q k abc m I a,b,c n I a,b,c k I a,b,c f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk 0 as m,n,k } Let G = max A a,b,c : a i 0, b l 0 and c j 0 and p,r and t be such that m i < p m i, n l < r n l and k j < t k j Thus we obtain the following: Q p Q r Q t p r t m=n=k= f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk m i n l k j ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk Q mi Q nl Q m=n= m I a,b,c n I a,b,c k= i l j Q mi Q nl Q a= b= c= f ((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk k I a,b,c = Q mi Q nl Q Q mk Q nl Q i 0 l 0 j 0 a= b= c= H a,b,c A a,b,c + (i 0<a i) (l 0<b l) (j 0<c j) G Q mi0 Q ni0 Q ki0 + Q mi Q nl Q Q mi Q nl Q H a,b,c A a,b,c H a,b,c A a,b,c (i 0<a i) (l 0<b l) (j 0<c j) G Q mi0 Q nl0 Q 0 + m i n l k j Q mi Q nl Q jj H a,b,c A a,b,c (i 0<a i) (l 0<b l) (j 0<c j) G Q mi0 Q nl0 Q 0

11 ) Triple Almost Lacunary Riesz definedby Orlicz Func39 µ nl + Q mi Q nl Q ( ) sup a i0 b l0 c j0 A a,b,c Q mi Q nl Q G Q mi0 Q nl0 Q 0 Q mi Q nl Q + ǫ Q mi Q nl Q H a,b,c (i 0<a i) (l 0<b l) (j 0<c j) H a,b,c (i 0<a i) (l 0<b l) (j 0<c j) = G Q mi0 Q nl0 Q 0 Q mi Q nl Q +V i V l V j ǫ G Q mi0 Q nl0 Q 0 Q mi Q nl Q +ǫh 3 Since Q mi Q nl Q as i,l,j approaches infinity, it follows that Q p Q r Q t p q m=n=k= uniformly in i,l and j Hence x t f((m+n+k)! x m+i,n+l,k+j ) /m+n+k p mnk = 0,,q,f,p µ nl Corollary 35 Let θ i,l,j = m i,n l,k j } be a triple lacunary sequence and be sequences of positive numbers If < lim ilj V ilj lim ilj sup ilj <, then for any Orlicz function f, µ nl γ ilj,q,f,p = χ 3 R λmi,q,f,p µ nl Definition 36 Let θ i,l,j = m i,n l,k j } be a triple lacunary sequence The triple number sequence x is said to be S P convergent to 0 provided χ 3 R ilj λmi µ nl that for every ǫ > 0, P lim sup ilj (m,n,k) I ilj : ilj } ((m+n+k)! x mnk ) /m+n+k, 0 ǫ = 0 In this case we write S χ 3 R λmi µ nl ilj P limx = 0

12 40 N Subramanian and A Esi Theorem 37 Let θ i,l,j = m i,n l,k j } be a triple lacunary sequence If I i,l,j I i,l,j, then the inclusion µ nl γ ilj,q S χ 3 R ilj λmi µ nl is strict and µ nl γ ilj,q P limx = S P limx = 0 χ 3 R ilj λmi µ nl Proof: Let (m,n,k) I } K Qilj (ǫ)= ilj :qmq q ((m+n+k)! x n k m+i,n+l,k+j ) /m+n+k, 0 ǫ (3) Suppose that x µ nl γ ilj,q Then for each i,l and j P lim ilj m I ilj n I ilj k I ilj ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 = 0 Since KQilj (ǫ) = m I ilj n I ilj m I ilj n I ilj k I ilj k I ilj ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 K Qilj (ǫ) for all i,l and j, we get P lim i,l,j λ iµ l γ j = 0 for each i,l and j This implies that x S χ 3 R ilj λmi µ nl

13 ) Triple Almost Lacunary Riesz definedby Orlicz Func4 µ nl To show that this inclusion is strict, let x = (x mnk ) be defined as (x mnk ) = 2 3 λ i µ l γ 4 m+n+k j Hi,l,j 0 (m+n+k)! 2 3 λ i µ l γ 4 m+n+k j Hi,l,j 0 (m+n+k)! λ i µ l γ 4 m+n+k j Hi,l,j (m+n+k)! 2 3 λ i µ l γ 4 m+n+k j Hi,l,j (m+n+k)! 0 λ i µ l γ 4 m+n+k j Hi,l,j λ i µ l γ 4 m+n+k j Hi,l,j λ i µ l γ 4 m+n+k j Hi,l,j ; 0 (m+n+k)! (m+n+k)! (m+n+k)! and q m = ;q n = ;q k = for all m,n and k Clearly, x is unbounded sequence For ǫ > 0 and for all i,l and j we have } (m,n,k) I ilj : q mq n q k ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ǫ ( λi µ l γ j (m+n+k)! 4 m+n+k H 4 m+n+k i,l,j Hi,l,j 4 m+n+k )/m+n+k Hi,l,j =P lim ilj 4 m+n+k(m+n+k)! Hi,l,j =0 Therefore x S χ 3 R λmi n l k j ilj with the P lim = 0 Also note that P lim q m q ilj H n q k ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ilj m I ilj n I ilj k I ilj ( =P λi µ l γ j (m+n+k)! 4 m+n+k H 4 m+n+k i,l,j Hi,l,j 4 m+n+k )/m+n+k Hi,l,j lim 2 ilj 4 m+n+k(m+n+k)! + Hi,l,j = 2 Hence x / µ nl γ ilj,q Theorem 38 Let I ilj I ilj If the following conditions hold, then χ 3 R λ m µ i nl γ ilj,q S µ χ 3 R ilj λmi µ nl

14 42 N Subramanian and A Esi and µ nl γ ilj,q P limx = S P limx = 0 µ χ 3 R ilj λi µ l γ j () 0 < µ < and 0 (2) < µ < and Proof: Let x = (x mnk ) be strongly µ nl γ ilj,q almost P conver- nj µ gent to the limit 0 Since ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 < < µ ((m+n+k)! x m+i,n+l,k+j ), 0 /m+n+k ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 for () and (2), for all i,l and j, we have q m q H n q k ilj m I ilj n I ilj k I ilj q m q H n q k ilj m I ilj n I ilj k I ilj ǫ KQilj (ǫ) µ ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 where K Qilj (ǫ) is as in (3) Taking limit i,l,j in both sides of the above inequality, we conclude that S P limx = 0 χ 3 R ilj λmi µ nl Definition 39 A triple sequence x = (x mnk ) is said to be Riesz lacunary of χ almost P convergent 0 if P lim i,l,j w ilj mnk (x) = 0, uniformly in i,l and j, where w ilj mnk (x) = wilj mnk = m I ilj n I ilj k I ilj ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 Definition 30 A triple sequence (x mnk ) is said to be Riesz lacunary χ almost statistically summable to 0 if for every ǫ > 0 the set } K ǫ = (i,l,j) N N N : w ilj ǫ mnk, 0

15 ) Triple Almost Lacunary Riesz definedby Orlicz Func43 µ nl has triple natural density zero, (ie) δ 3 (K ǫ ) = 0 In this we write χ 3Rλmiµnlγ ilj st 2 P limx = 0 That is, for every ǫ > 0, P lim rst rst i r,l s,j t : w ilj mnk, 0 ǫ} = 0,uniformly in i,l and j Theorem 3 Let I ilj I ilj and ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 M for all m,n,k N and for each i,l and j Let x = (x mnk ) be S χ 3 R λmi µ nl ilj P limx = 0 Let } K Qilj (ǫ) = (m,n,k) I ilj : q mq n q k ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 ǫ Then w ilj mnk, 0 = λ i µ l γ j M KQilj (ǫ) +ǫ m I ilj n I ilj, 0 ((m+n+k)! x m+i,n+l,k+j ) /m+n+k k I ilj ((m+n+k)! x m+i,n+l,k+j ) /m+n+k, 0 m I ilj n I ilj k I ilj for each i,l and j, which implies that P lim i,l,j w ilj mnk (x) = 0, uniformly i,l and j Hence, St 2 P lim ilj w ilj mnk = 0 uniformly in i,l,j Hence χ 3 R λi µ l γ ilj j st 2 P limx = ( 0 To see that the converse is not true, consider the triple lacunary sequence θ ilj 2 i 3 l 4 j )},q m =,q n =,q k = for all m,n and k, and the triple sequence x = (x mnk ) defined by x mnk = ( )m+n+k (m+n+k)! for all m,n and k Competing Interests: The authors declare that there is not any conflict of interests regarding the publication of this manuscript References T Apostol, Mathematical Analysis Addison-Wesley, London, 978

16 44 N Subramanian and A Esi 2 A Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions Research and Reviews: Discrete Mathematical Structures (2), 6-25, (204) 3 A Esi and M Necdet Catalbas, Almost convergence of triple sequences Global Journal of Mathematical Analysis 2(), 6-0, (204) 4 A Esi and E Savas, On lacunary statistically convergent triple sequences in probabilistic normed space Appl Math and Inf Sci 9(5), , (205) 5 A Esi, Statistical convergence of triple sequences in topological groups Annals of the University of Craiova, Mathematics and Computer Science Series 40(), 29-33, (203) 6 E Savas and A Esi, Statistical convergence of triple sequences on probabilistic normed space Annals of the University of Craiova, Mathematics and Computer Science Series 39(2), , (202) 7 G H Hardy, On the convergence of certain multiple series Proc Camb Phil Soc 9, 86-95, (97) 8 N Subramanian, C Priya and N Saivaraju, The χ 2I of real numbers over Musielak metric space Southeast Asian Bulletin of Mathematics 39(), 33-48, (205) 9 Deepmala, N Subramanian and V N Mishra, Double almost (λ mµ n ) in χ 2 Riesz space Southeast Asian Bulletin of Mathematics 35, -, (206) 0 Deepmala, L N Mishra and N Subramanian, Characterization of some Lacunary χ 2 A uv convergence of order α with p metric defined by mn sequence of moduli Musielak Appl Math Inf Sci Lett 4(3), 9-26, (206) A Sahiner, M Gurdal and F K Duden, Triple sequences and their statistical convergence Selcuk J Appl Math 8(2), 49-55, (2007) 2 N Subramanian and A Esi, Some new semi-normed triple sequence spaces defined by a sequence of moduli Journal of Analysis & Number Theory 3(2), 79-88, (205) 3 T V G Shri Prakash, M Chandramouleeswaran and N Subramanian, The Random of Lacunary statistical on Γ 3 over metric spaces defined by Musielak Orlicz functions Modern Applied Science 0(), 7-83, (206) 4 T V G Shri Prakash, M Chandramouleeswaran and N Subramanian, Lacunary Triple sequence Γ 3 of Fibonacci numbers over probabilistic p metric spaces International Organization of Scientific Research 2(), 0-6 (206) 5 H Nakano, Concave modulars Journal of the Mathematical Society of Japan 5, 29-49, (953) Nagarajan Subramanian Department of Mathematics, SASTRA University, Thanjavur-63 40, India address: nsmaths@yahoocom and Ayhan Esi Department of Mathematics, Adıyaman University, 02040, Adiyaman, Turkey address: aesi23@hotmailcom