Introduction to generalised Cesaro convergence II

TL;DR

This paper extends Cesaro convergence to remainder Cesaro summation, broadening its applicability and providing new insights into the Gamma function and functional equations through geometric and dilation-invariance properties.

Contribution

It introduces a new form of Cesaro convergence called remainder Cesaro summation, expanding the scope of problems it can address and offering a novel geometric perspective.

Findings

New definition of the Gamma function via Cesaro summation

Cesaro convergence is invariant under dilation and scaling

Method explains structure of functional equations

Abstract

In this second of three introductory papers, we extend the notion of generalised Cesaro summation/convergence to the more natural setting of what we call remainder Cesaro summation/convergence. This greatly expands the range of problems susceptible to Cesaro methods and introduces the geometric location of summands as a critical consideration. We also show that geometric generalised Cesaro convergence is invariant under dilation and scaling. We present a number of calculations illustrating the utility of these developments. In particular we introduce a new, more natural definition of the classical Gamma function using remainder Cesaro summation/products, and show that many its key properties - both basic and advanced - fall out directly and intuitively from this Cesaro definition and its geometric and dilation-invariance properties. We also consider other examples and show how Cesaro methodology explains the common structure of many well-known functional equations.