Product theorems for certain summability methods in non-Archimedean fields (Q1881469)

Let \(A=(a_{nk})\) be a matrix and \(x=\{x_{k}\}\) a sequence with entries in a non-Archimedean field. The sequence \(A(x)=\{(Ax)_{n}\}\), where \[ (Ax)_{n}=\sum^{\infty}_{k=0}a_{nk}x_{k}, \qquad n=0,1,2,\dots \] is called \(A\)-transform of \(x=\{x_{k}\}\). If \(\lim_{n\rightarrow\infty}(Ax)_{n}=l\), it is said that \(x=\{x_{k}\}\) is \(A\)-summable to \(l\). If \(\lim_{n\rightarrow\infty}(Ax)_{n}=l\), whenever \(\lim_{k\rightarrow\infty}x_{k}=l\), then it is said that the method \(A\) is regular. Given two regular methods \(A\) and \(B\), a result of the form ``\(A(x)\) converges implies \(A(B(x))\) converges to the same limit'' is called a product theorem. The main aim of the paper under review is to prove two product theorems that involve \(M\) and \((N,p_{n})\) summability methods.