Introduction to generalised Cesaro convergence II
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918464787054592 |
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| author | Stone, Richard |
| author_facet | Stone, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21947 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Introduction to generalised Cesaro convergence II Stone, Richard General Mathematics 40A10 (Primary), 40A05 (Primary), 33B15 (Primary), 11M35 (Secondary) 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. |
| title | Introduction to generalised Cesaro convergence II |
| topic | General Mathematics 40A10 (Primary), 40A05 (Primary), 33B15 (Primary), 11M35 (Secondary) |
| url | https://arxiv.org/abs/2604.21947 |