A positive sequence (v_n) is said to be almost increasing if there exists a positive increasing sequence (c_n) and two positive constants K and L such that Kc_n ≤ v_n ≤ Lc_n (see [1]). A sequence (y_n) is said to be δ-quasi-monotone, if y_n → 0, y_n > 0 ultimately and Δy_n ≥ −δ_n, where Δy_n=y_n – y_n+1 and δ = (δ_n) is a sequence of positive numbers (see [2]). Let ∑a_n be a given infinite series with partial sums (s_n). By (u_n) and (t_n) we denote the n-th (C, 1) means of the sequences (s_n) and (na_n), respectively. The series ∑a_n is said to be |C, 1|_k summable, k ≥ 1, if (see [6], [8])

Let (p_n) be a sequence of positive numbers such that

The sequence-to-sequence transformation

defines the sequence (z_n) of the Riesz mean of the sequence (s_n), generated by the sequence of coefficients (p_n) (see [7]). The series ∑a_n is said to be |N̄, p_n|_k summable, k ≥ 1, if (see [3])

where

Let A = (a_nv) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (s_n) to As = (A_n(s)), where

The series ∑a_n is said to be |A, p_n|_k summable, k ≥ 1, if (see [9])

where

When we take anv=pvPn, then |A, p_n|_k summability is the same as |N̄, p_n|_k summability. Also, when we take anv=pvPn and p_n = 1 for all values of n, |A, p_n|_k reduces to |C, 1|_k summability.

Let A = (a_nv) be a normal matrix. Lower semimatrices Ā = (ā_nv) and  = (â_nv) are defined as follows:

and

Ā and  are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we write

and

In [4, 5], the following theorem dealing with |N̄, p_n|_k summability factors of infinite series has been proved by Bor.

Theorem 2.1

Let (X_n) be an almost increasing sequence such that |ΔX_n| = O(X_n/n) and λ_n → 0 as n → ∞. Suppose that there exists a sequence of numbers (A_n) such that it is δ-quasi-monotone with ∑nX_nδ_n < ∞, ∑A_nX_n is convergent and |Δλ_n| ≤ |A_n| for all n. If

and

then the series ∑a_nλ_n is |N̄, p_n|_k summable, k ≥ 1.

The aim of this paper is to prove following more general theorem dealing with |A, p_n|_k summability.

Theorem 3.1

Let A = (a_nv) be a positive normal matrix such that

If all conditions of Theorem 2.1 are satisfied, then the series ∑a_nλ_n is |A, p_n|_k summable, k ≥ 1.

Lemma 3.2.([4])

Under the conditions of Theorem 3.1, we have

Lemma 3.3.([5])

Let (X_n) be an almost increasing sequence such that n |ΔX_n |= O (X_n). If (A_n) is a δ-quasi monotone with ∑nX_nδ_n < ∞, and ∑A_nX_n is convergent, then

Let (M_n) denotes A-transform of the series ∑a_nλ_n. Then, by (1.10) and (1.11), we have

Applying Abel’s transformation to above sum, we get

To prove Theorem 3.1, we will show that

First, by using (2.3), (3.3) and (3.4), we have

Now, as in M_n,1, we have

Since

by (1.8) and (1.9), we have

by using (1.8), (3.1) and (3.2). Hence, we get

Now, using (3.2) and (4.1), we obtain

then

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Also, we have

By (1.8), (1.9), (3.1) and (3.2), we obtain

Thus, using (1.8) and (3.1), we have

then we get

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.3. Again, operating Hölder’s inequality, we have

by (2.1), (2.2), (3.3) and (3.4). This completes the proof of Theorem 3.1.

If we take anv=pvPn in this theorem, then we get Theorem 2.1. If we take anv=pvPn and p_n = 1 for all values of n, then we get a result for |C, 1|_k summability.