The λ+r(μ)-statistical convergence

Abstract

Let $\lambda =({\lambda }_n{)}_{n\ge 1}$ be a nondecreasing sequence of positive numbers tending to infinity such that ${\lambda }_1=1$ and ${\lambda }_{n+1}\le {\lambda }_n+1$ for all $n$, and let $I_n=[n-{\lambda }_n+1,n]$ for $n=1,2,\dots$. Then for any given nonzero sequence $\mu$, we define by ${\Delta }^{+}\left(\mu \right)$ the operator that generalizes the operator of the first difference and is defined by ${\Delta }^{+}\left(\mu \right)x_k={\mu }_k(x_k-x_{k+1})$. In this article, for any given integer $r\ge 1$, we deal with the ${\lambda }^{+r}\left(\mu \right)$ -statistical convergence that generalizes in a certain sense the well-known ${\lambda }_E^r$-statistical convergence. The main results consist in determining sets of sequences $\chi$ and ${\chi }^{\prime }$ of the form $s_{\xi }^0$ satisfying $\chi \subset [V,\lambda {]}_0({\Delta }^{+r}\left(\mu \right))\subset {\chi }^{\prime }$ and sets $\kappa$ and ${\kappa }^{\prime }$ of the form $s_{\xi }$ satisfying $\kappa \le [V,\lambda {]}_{\infty }({\lambda }^{+r}\left(\mu \right))\le {\kappa }^{\prime }$. This study is justified since the infinite matrix associated with the operator ${\Delta }^{+r}\left(\mu \right)$ cannot be explicitly calculated for all $r$.