THE -TRANSLATIVITY OF ABEL-TYPE MATRIX

Lemma introduced the Abel-type matrix Aα,t defined by ank = ( k+α k ) tk+1 n (1− tn)α+1, where α>−1, 0< tn < 1, for all n, and limtn = 1; and studied it as mappings into . In this paper, we extend our study of this matrix and investigate its translativity in the setting.

1. Background.In [1], Borwein proved that the Abel-type power series method of summability denoted by A α (α > −1), is translative in the ordinary summability (c-c) setting.So, it natural to ask if there is a theory in the -setting that parallel the theory of A α in the c-c setting.The answer is affirmative, and have provided the present study.

Basic notation and definitions.
Let A = (a nk ) be an infinite matrix defining a sequence to a sequence summability transformation given by where (Ax) n denotes the nth term of the image sequence Ax.Let y be a complex number sequence.Throughout this paper, we use the following basic notations and definitions: (i) c = {the set of all convergent complex number seqences}, (ii) = y : If X and Y are sets of complex number sequences, then the matrix A is called an X-Y matrix if the image Au of u under the transformation A is in Y whenever u is in X. Definition 2. The summability matrix A is said to be -translative for the sequence u in (A) provided that each of the sequences T u and S u is in (A), where T u = {u 1 ,u 2 ,u 3 ,...} and S u = {0,u 0 ,u 1 ,...}.

The main results
Proposition 1.Every -A α,t matrix is -translative for each sequence x ∈ .Theorem 1.Every -A α,t matrix is -translative for those sequences x ∈ (A α,t ) for which {x k /k} ∈ , k = 1, 2, 3,... .Proof.Suppose that x is a sequence in (A α,t ) for which {x k /k} ∈ .We show that (1) T x ∈ (A α,t ), and (2) S x ∈ (A α,l ), where T x and S x are as defined in Definition 2. Let us first show that (1) holds.Note that where and The use of the triangle inequality in equation (3.1) is legitimate as the radii of convergence of the two power series are at least 1.Now if we show that both A and B are in , then (1) holds.But the conditions that A ∈ and B ∈ follow easily from the hypotheses that x ∈ (A α,t ) and {x k /k} ∈ , respectively.Next, we show that (2) holds as follows.We have where and The use of the triangle inequality in (3.4) is justified as above.If we show that E and F are in , then (2) holds.But the hypothesis that x ∈ (A α,t ) and {x k /k} ∈ implies that both E and F are in , respectively, and hence the theorem follows.
Here, we remark that a sequence x defined by x k = (−1) k /k is one of the sequences which satisfies the condition of Theorem 1.

Proof. Let x ∈ c(A α ) ∩ (A α,t ). We show that (1) T x ∈ (A α,t ), (2) S x ∈ (A α,t ).
First, let us show that (1) holds.Since the case α = 0 was already proved by J. Fridy in [2], here we only consider the case −1 < α < 0. Note that where and Now if we show that both A and B are in , then (1) holds.But the condition that A ∈ follows from the hypothesis that x ∈ (A α,t ) and B ∈ are shown as follows.Observe that where and By [3, Thm.1], the hypothesis that A α,t -implies that C ∈ , hence there remains only to show D ∈ to show that B ∈ .Observe that (3.13) The interchanging of the integral and the summation is legitimate as the radius of convergence of the power series is at least 1 by [3, Prop.1], and hence the power series converges absolutely and uniformly for 0 ≤ t ≤ t n .Now we let Then, we have and the hypothesis that x ∈ c(A α ) implies that lim (3.17) We also have Now equations (3.17) and (3.18) yield that and hence So, we have (3.21) By [3, Thm.1], the hypothesis that A α,t -implies that (1 − t) α+1 ∈ , and hence D ∈ .Next we show that (2) holds.We have (3.22)where and Now if we show that both E and F are in , then (2) follows.But the hypothesis that x ∈ (A α,t ) implies that E ∈ , and F ∈ follows using the same technique used in showing D ∈ in (1) in the proof of Theorem 2. Hence the theorem is proved.
Corollary 1. Suppose that −1 ≤ α ≤ 0; then every -A α,t matrix is -translative for the class of all sequence x whose partial sum is bounded.
Proof.By [3, Thm.8], x is in (A α,t ) and it is easy to see that x is also in c(A α ).Hence the assertion follows by Theorem 2.
Corollary 2. Suppose that −1 ≤ α ≤ 0; then every -A α,t matrix is -translative for the unbounded sequence x defined by ), the corollary easily follows by Theorem 2.
Proof.Suppose that x ∈ c(A α ) ∩ (A α,t ).To prove the theorem, we need to show that both A α,t T x and A α,t S x are in .We have and by referring to the proof of Theorem 1, we can express where and Now if we show that both A and B are in , then A α,t T x is in .The condition that A ∈ follows easily since A α,t is an -matrix and x ∈ (A α,t ).The condition that B ∈ is shown as follows.Using the triangle inequality, we have where Now since (1 − t) ∈ is given and A α,t is an -matrix implies that (1 − t) α+1 is in by [3, Thm.1], it follows that D ∈ .Also using the same techniques as in the proof of (2) of Theorem 2, we can easily show that A α,t S x ∈ and hence the assertion follows.
.32)We have B ∈ , if we show that both C and D are in .The condition that C ∈ follows easily as (1 − t) α+1 ∈ by [3, Thm.1], and D ∈ can be shown as follows.Note that following exactly the same steps as in the proof of (1) of Theorem 2, we can easily show that