On absolute summability factors (Q5917797)

The author proves that if the series \(\sum a_n\) is \([\overline N, p_n, \delta ]_k\)-bounded and the sequences \((\lambda_n)\) and \((p_n)\) satisfy a certain set of conditions and if \[ \sum^\infty_{n = v + 1} (P_n/p_n)^{\delta k - 1} (1/P_{n - 1}) = O \bigl( (P_v/p_v)^{ \delta k} (1/P_v) \bigr) \] then the series \(\sum a_n P_n \lambda_n\) is \([\overline N, p_n, \delta] _k\)-summable for \(k \geq 1\) and \(\delta \geq 0\). This result generalizes the author's previous result on \([\overline N, p_n, \delta]_k\)-summability factors of infinite series \(\sum a_n P_n \lambda_n\).