Some new characterizations on absolute summability factors

Let A and B be two summability methods and let ε=(ε_n ) be a sequence of complex numbers. If ∑〖ε_n x_n 〗 is summable B whenever ∑x_n is summable A, then the sequence ε is said to be a summability factor of type (A,B), and it is denoted by ε∈(A,B). The summability method|N ̅,p_n | and the summability method |C,a|_k are generally independent of each other. Therefore, it is natural to search for suitable summability factors of the type (|C,a|_k,|N ̅,p_n |). In this direction, some important studies made on summability factors have been given. The |N ̅,p_n,θ_n |_k summability method has been extended to the |A_f,θ_n |_k summability method by using the multiplicative matrix A=(a_nv ) and the |C,a|_k absolute Cesàro summability method has been extended to the interval a≥-1 by the |C,-1|_k method. In order to expand the studies on summability factors made with these methods, the summability factors related to |C,-1|_k and |A_f,θ_n |_k summability methods will be characterized. Thus, known results will be expanded.