In the present paper, absolute matrix summability of infinite series is studied. A new theorem concerning absolute matrix summability factors, which generalizes a known theorem dealing with absolute Riesz summability factors of infinite series, is proved using almost increasing and δ-quasi-monotone sequences. Also, a result dealing with absolute Cesàro summability is given.
A positive sequence (vn) is said to be almost increasing if there exists a positive increasing sequence (cn) and two positive constants K and L such that Kcn ≤ vn ≤ Lcn (see ). A sequence (yn) is said to be δ-quasi-monotone, if yn → 0, yn > 0 ultimately and Δyn ≥ −δn, where Δyn=yn – yn+1 and δ = (δn) is a sequence of positive numbers (see ). Let ∑an be a given infinite series with partial sums (sn). By (un) and (tn) we denote the n-th (C, 1) means of the sequences (sn) and (nan), respectively. The series ∑an is said to be |C, 1|k summable, k ≥ 1, if (see , )
Let (pn) be a sequence of positive numbers such that
The sequence-to-sequence transformation
defines the sequence (zn) of the Riesz mean of the sequence (sn), generated by the sequence of coefficients (pn) (see ). The series ∑an is said to be |N̄, pn|k summable, k ≥ 1, if (see )
Let A = (anv) 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 = (sn) to As = (An(s)), where
The series ∑an is said to be |A, pn|k summable, k ≥ 1, if (see )
When we take , then |A, pn|k summability is the same as |N̄, pn|k summability. Also, when we take and pn = 1 for all values of n, |A, pn|k reduces to |C, 1|k summability.
Let A = (anv) be a normal matrix. Lower semimatrices Ā = (ānv) and Â = (ânv) are defined as follows:
Ā and Â are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we write
In [4, 5], the following theorem dealing with |N̄, pn|k summability factors of infinite series has been proved by Bor.
Let (Xn) be an almost increasing sequence such that |ΔXn| = O(Xn/n) and λn → 0 as n → ∞. Suppose that there exists a sequence of numbers (An) such that it is δ-quasi-monotone with ∑nXnδn < ∞, ∑AnXn is convergent and |Δλn| ≤ |An| for all n. If
then the series ∑anλn is |N̄, pn|k summable, k ≥ 1.