Absolute varphi- vert C, alpha, beta; deltavert _{k} summability of infinite series

Smita Sonker,Alka Munjal

doi:10.1186/s13660-017-1445-5

Abstract

In this paper, we established a generalized theorem on a minimal set of sufficient conditions for absolute summability factors by applying a sequence of a wider class (quasi-power increasing sequence) and the absolute Cesàro varphi-vert C, alpha, beta; delta vert _{k} summability for an infinite series. We further obtained well-known applications of the above theorem as corollaries, under suitable conditions.

Highlights

Let ∞ n=an be an infinite series with sequence of partial sums {sn}and the nth sequence to sequence transformation of {sn} be given by un s.t. ∞ un = unksk. ( ) k=Before discussing φ – |C, α, β; δ|k summability, let us introduce some well-known basic summabilities which are helpful in understanding the φ – |C, α, β; δ|k summability.Definition The series an is said to be absolute summable, if lim n→∞
In, Bor [ ] generalized the |C, α|k summability factor to the |C, α, β; δ|k summability of an infinite series and in [ ], he discussed a general class of power increasing sequences and absolute Riesz summability factors of an infinite series
In [ ], Bor applied |C, α, γ ; β|k summability to obtain the sufficient conditions for an infinite series to be absolute summable

Summary

Introduction

The nth sequence to sequence transformation (mean) of {sn} be given by un s.t. un = unksk. Bor gave a number of theorems on absolute summability. In , Bor [ ] generalized the |C, α|k summability factor to the |C, α, β; δ|k summability of an infinite series and in [ ], he discussed a general class of power increasing sequences and absolute Riesz summability factors of an infinite series. In [ ], Bor applied |C, α, γ ; β|k summability to obtain the sufficient conditions for an infinite series to be absolute summable. Bor [ ] gave a new application of quasi-power increasing sequence by applying absolute Cesáro φ – |C, α|k summability for an infinity series. In , Sonker and Munjal [ ] determined a theorem on generalized absolute Cesáro summability with the sufficient conditions for an infinite series and in [ ], they used the concept of triangle matrices for obtaining the minimal set of sufficient conditions of an infinite series to be bounded.

Main results

Conclusion