Introduction to generalised Cesaro convergence II

Richard Stone

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arXiv · 2604.21947

Introduction to generalised Cesaro convergence II

Richard Stone

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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.

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Richard Stone. 2026-04-22. Introduction to generalised Cesaro convergence II. https://arxiv.org/abs/2604.21947

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