Some Results of Generalized Weighted Norlund-Euler-<img src=image/13428701_01.gif> Statistical Convergence in Non-Archimedean Fields

Abstract

Non-Archimedean analysis is the study of fields that satisfy the stronger triangular inequality, also known as ultrametric property. The theory of summability has many uses throughout analysis and applied mathematics. The origin of summability methods developed with the study of convergent and divergent series by Euler, Gauss, Cauchy and Abel. There is a good number of special methods of summability such as Abel, Borel, Euler, Taylor, Norlund, Hausdroff in classical Analysis. Norlund, Euler, Taylor and weighted mean methods in Non-Archimedan Analysis have been investigated in detail by Natarajan and Srinivasan. Schoenberg developed some basic properties of statistical convergence and also studied the concept as a summability method. The relationship between the summability theory and statistical convergence has been introduced by Schoenberg. The concept of weighted statistical convergence and its relations of statistical summability were developed by Karakaya and Chishti. Srinivasan introduced some summability methods namely y-method, Norlund method and Weighted mean method in p-adic Fields. The main objective of this work is to explore some important results on statistical convergence and its related concepts in Non-Archimedean fields using summability methods. In this article, Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence, generalized weighted summability using Norlund-Euler-<img src=image/13428701_02.gif> method in an Ultrametric field are defined. The relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and Statistical Norlund-Euler-<img src=image/13428701_02.gif> summability has been extended to non-Archidemean fields. The notion of Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and inclusion results of Norlund-Euler statistical convergent sequence has been characterized. Further the relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence of order α & β has been established.