Parametric Summability and Its Applications

TL;DR

This paper introduces a new framework for summability based on power double sequences, generalizing classical double sequences and power series, and explores their properties and applications.

Contribution

It defines novel power double sequences, relates their summability to classical double sequences, and extends existing summability results to these new constructs.

Findings

Established relations between power double sequences and classical double sequences.

Extended summability results to the context of power double sequences.

Assessed the usefulness of various generalizations of double sequences.

Abstract

In this paper we study summability based on double sequences of complex constants as it is defined in "Linear Operators, General Theory" by N. Dunford and J. T. Schwartz. We define "power double sequences" or infinite "power matrices" as certain generalizations of double sequences and power series. We relate the summability and boundedness of the power double sequences to the summability and boundedness of the double sequence \textit{A}. While others do investigate "power matrices" their definitions, as far as we were able to find, differ from our definitions. Using these definitions we extend some summability results for double sequences of constants to our power double sequences. We investigate also other possible generalizations of double sequences and assess their usefulness in summability study.