\(q\)-Cesàro matrix and \(q\)-statistical convergence
Let \(A=(a_{nk})\) be an infinite matrix, and let \(x=(x_k)\) be a sequence of real numbers. The \(A\)-transform of \(x\) is given by \[ (Ax)_n=\sum_{k=0}^\infty a_{nk}x_k \] provided that the series converges for each \(n\geq 0\). The matrix \(A\) is said to be conservative (regular) if the convergence of \(x\) implies the convergence of \((Ax)_n\) (\(\lim x_n=\lim (Ax)_n\)). A very important example of a regular matrix is the Cesàro matrix associated with the Cesàro summability method. The authors give the following generalisation of the Cesàro matrix. Let \(S_q=(a_{nk}(q))\) be an infinite matrix such that \(a_{nk}(q)=0\) if \(k>n\) and \(s_{nk}(1)=1\) for \(k\leq n\). Then \(S_q\) is called the \(q\)-analog of the summation matrix generated by \(a_{nk}(q)\). Using the standard procedure of defining the Cesàro matrix of order \(\alpha\in\mathbb N\), the authors define a \(q\)-Cesàro matrix of order \(\alpha\). They give a precise formula for the \(q\)-Cesàro matrix of order one. Then they focus on the special case when \(a_{nk}(q)=q^k\) for \(k\leq n\). The matrix given by \(q^k\), \(k\leq n\), is conservative for \(q\in\mathbb R\) and regular for \(q>1\). The associated summation methods with \(q_1>q_2>0\) are equivalent, but the classical Cesàro method is stronger than that associated with \(q>1\). Finally, the authors define \(q\)-statistical convergence using the \(q\)-Cesàro matrix, and they prove that this notion differs from ordinary statistical convergence.
scientific article; zbMATH DE number 5930993
scientific article