Characterization on some absolute summability factors of infinite series (Q1970286)

In a previous paper, the author [Proc. Am. Math. Soc. 115, No. 2, 313-317 (1992; Zbl 0756.40006)] defined a series \(\Sigma a_n\) to be summable \(|N,p_n, \varphi_n|_k\), \(k\geq 1\), if \(\sum^\infty_{n=1} \varphi_n^{k-1} |T_n-T_{n-1} |^k <\infty\), where \(T_n=P_n^{-1} \sum^n_{i=0} p_is_i\). By placing certain restrictions on the sequences \(\{p_n\}\), \(\{q_n\}\), \(\{\alpha_n\}\), and \(\{\beta_n\}\), the author, in this paper, obtains necessary and sufficient conditions for the \(|N,q_n, \beta_n |_k\)-summability of \(\Sigma a_n\) to imply the \(|N,p_n, \alpha_n |_k\)-summability of \(\Sigma a_n\varepsilon_n\). Editorial comment: Apart from a few lines added at the end, the paper is identical to the author's paper [Ganita Sandesh 11, No. 1, 1--6 (1997; Zbl 1186.40010)] which is not mentioned here.