\(q\)-Cesàro matrix and \(q\)-statistical convergence
type: Item
instance of: scholarly article
title: \(q\)-Cesàro matrix and \(q\)-statistical convergence
zbMATH Open document ID: 1229.40008
author: Hüseyin Aktuğlu
author: Şerife Bekar
published in: Journal of Computational and Applied Mathematics
publication date: 2011-07-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 40C05
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 40A35
zbMATH DE Number: 5930993
zbMATH Keywords: Cesàro matrices
zbMATH Keywords: Cesàro summability
zbMATH Keywords: \(q\)-integers
zbMATH Keywords: statistical convergence
reviewed by: Szymon Glab
MaRDI profile type: Publication
full work available at URL: j.cam.2010.08.018
OpenAlex ID: W1564557590
cites work: Q4520851
cites work: Convergence of generalized Bernstein polynomials
cites work: Q3365622
cites work: Densities and summability
cites work: Sur la convergence statistique
cites work: ON STATISTICAL CONVERGENCE
DOI: 10.1016/J.CAM.2010.08.018
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description: scientific article; zbMATH DE number 5930993