Inclusion theorems for strong and absolute Nörlund summability (Q1911902)

Let \(\{p_n\}\), \(\{q_n\}\), \(\{u_n\}\) be sequences, \[ P_n:= \sum^n_{k= 0} p_k,\;Q_n:= \sum^n_{k= 0} q_k,\;w_n:= \sum^n_{k= 0} u_{n- k} q_k,\;r_n:= \sum^n_{k= 0} p_{n- k} q_k. \] A sequence \(\{s_n\}\) is defined to be strongly generalized Nörlund summable, with index \(\lambda> 0\), to \(s\), written \(s_n\to s[N, p, q]_\lambda\) if \[ \sum^N_{n= 0} |\Delta r_n||t^{(\Delta p, q)}_n- s|^\lambda= o(|r_N|),\quad \text{where } t^{(\Delta p, q)}_n:= \sum^n_{k= 0} (\Delta p_{n- k}) q_k s_k/ \Delta r_n, \] \(\Delta\) denoting the forward difference operator. If the transformed sequence \(\{t^{(\Delta p, q)}_n\}\) is of bounded variation, then \(\{s_n\}\) is said to be absolutely generalized summable, written \(\{s_n\}\in |N, p, q|\). Let \[ \mathbb{M}= \{p= \{p_n\}\mid p_n> 0\text{ and } p_{n+ 1}/ p_n\leq p_{n+ 2}/ p_{n+ 1}\text{ for each } n\geq 0\}. \] Theorem 1. Let \(\{p_n\}\in \mathbb{M}\), \(\{u_n\}\) nondecreasing, \(\{p_n\}\) increasing, all sequences positive, with \(\lim r_n= \infty\). Then \(s_n\to s[N, p, q]_\lambda\) implies \(s_n\to s[N, u, q]_\lambda\) iff \[ \sum^n_{k= 0} u_{n- k} c^{(1)}_k= o(w_0),\text{ where } c^{(1)}_n:= \sum^n_{k= 0} c_k, \{c_n\}\text{ defined by } \sum c_n x^n= (\sum p_n x^n)^{- 1}. \] Sufficient conditions are also obtained for \([\overline N, Q]_\lambda\) to imply \([N, p, Q]_\lambda\) and \(|\overline N, q|\) to imply \(|N, p, q|\).