Inclusion theorems for generalised Nörlund summability methods (Q1891546)

A sequence \(\{s_ n\}\) is generalized Nörlund summable to \(s\) with respect to given sequences \(\{p_ n\}\) and \(\{q_ n\}\) (real or complex), written \(s_ n \to s(N,p,q)\), if \(\sum^ n_{i = 0} p_{n - i} q_ i s_ i/ \sum^ n_{j = 0} p_{n - j} q_ j \to s\). \((N,p,q)\)-summability becomes Nörlund summability when \(q_ n = 1\), \(n \geq 0\), and is \((\overline N,q)\)-summability when \(p_ n = 1\), \(n \geq 0\). Necessary and sufficient conditions are presented for \(s_ n \to s(N,p,q)\) to imply \(s_ n \to s(N,u,q)\), where \(\{u_ n\}\) is positive and non-decreasing. Conditions tantamount to the equivalence of \((N,p,q)\) and \((\overline N,q)\)-summability are derived, and circumstances under which \((N,p,q)\) or \((\overline N,q)\)-summability is the same as the usual convergence of \(s_ n \to s\) are established.