On the \(|\overline{N},p_n|_k\) summability factors for infinite series (Q1182597)

Let \(\sum a_n\) be a given infinite series with the sequences of partial sums \(\{s_n\}\). Let \(\{p_n\}\) be a sequence of positive real constants such that \[ P_n=\sum_{\nu=0}^ n p_\nu\to\infty\quad\text{as } n\to\infty \qquad (P_{-i}=p_{-i}=0,\;i\geq 1). \] The sequence-to-sequence transformation \(t_n=\frac{1}{P_n}\sum_{\nu=0}^n p_\nu s_\nu\) defines the sequence \(\{t_n\}\) of the \((\overline{N},p_n)\)-mean of the sequence \(\{s_n\}\), generated by the sequence of coefficients \(\{p_n\}\). The series \(\sum a_n\) is summable \(|\overline{N},p_n|_k\), \(k\geq 1\), if \[ \sum_{n=1}^\infty (P_n/p_n)^{k-1}| t_n-t_{n- 1}|^ k<\infty. \] The author generalizes his theorem [Proc. Indian Acad. Sci., Math. Sci. 98, No. 1, 53--57 (1988; Zbl 0694.40012)] and proves the following interesting theorem: Let \(\sum a_n\) be bounded \([\overline{N},p_n]_ k\). If the sequences \(\{p_n\}\), \(\{\beta_n\}\) and \(\{\lambda_n\}\) such that the conditions (i) \(\frac{1}{n}=o(p_n)\); (ii) \(|\Delta\lambda_n|\leq\beta_n\); (iii) \(\beta_n\to 0\) as \(n\to\infty\); (iv) \(\sum_{n=1}^\infty nP_n|\Delta\beta_n|<\infty\); (v) \(P_n|\lambda_ n|- o(1)\) as \(n\to\infty\); then the series \(\sum a_n\lambda_n\) is summable \(|\overline{N},p_n|_k\), \(k\geq 1\).