On certain families of generalized Nörlund methods and power series methods (Q1307251)

A generalized Nörlund method is generated by a sequence \(\{r_n\}\), where \(r_n=(a*b)_n:=\sum^n_{k=0}a_{n-k}b_k\), and is denoted by \((N,a, b)\). A sequence \(\{s_n]\) is summable to \(s\) by a power series summability method \(J_q\), if the power series \(q(x)=\sum^\infty_{n=0}q_nx^n\) has a radius of convergence of \(1\), \(q_s(x): =\sum^\infty_{n=0} s_nq_nx^n\) converges for \(|x|<1\) and \((1/q(x)) \sum^\infty_{n=0} s_nq_nx^n\to s\) as \(x\to 1-\). The extended power series method \(J_q^*\) has the property that \(q_s(x)\) has a positive radius of convergence, can be holomorphically extended along \((0,1)\) and \(q_s(x)/q(x)\to s\) as \(x\to 1-\). The authors obtain inclusion relations for \(J_q\), \(J_q^*\), \(J_r\cdot (N,a,q)\), and \((N,a,q)\cdot J_r\). A family of summability methods \((A_\alpha\}\) is called a Cesàro-type family if the methods \(A_\alpha\) and \(A_\alpha'\), for any \(\alpha_0<\alpha <\alpha'\), are related by the equation \(\eta_n^{\alpha'}=(1/b_n^{\alpha'})\sum^n_{k=0}A_{n-k}^{\alpha'- \alpha-1} b_k^\alpha \eta_k^\alpha\). The authors obtain inclusion and Tauberian theorems for certain matrices \(A_\alpha\). They also obtain inclusion and Tauberian theorems for generalized Nörlund matrices of the form \((N,p^{\alpha \beta},q)\), where \(p_n^{\alpha \beta}:=(p^{\alpha,\beta-1}*p)_n\).