Formal Calculation

Ji Peng

doi:10.20944/preprints202305.0311.v1

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preprints.org > computer science and mathematics > discrete mathematics and combinatorics > doi: 10.20944/preprints202305.0311.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Formal Calculation

Ji Peng

Version 1 : Received: 3 May 2023 / Approved: 5 May 2023 / Online: 5 May 2023 (06:23:10 CEST)
Version 2 : Received: 1 August 2023 / Approved: 1 August 2023 / Online: 2 August 2023 (10:11:58 CEST)
Version 3 : Received: 5 October 2023 / Approved: 5 October 2023 / Online: 7 October 2023 (03:26:10 CEST)

How to cite: Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v1 Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v1

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MDPI and ACS Style

Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v1

APA Style

Peng, J. (2023). Formal Calculation. Preprints. https://doi.org/10.20944/preprints202305.0311.v1

Chicago/Turabian Style

Peng, J. 2023 "Formal Calculation" Preprints. https://doi.org/10.20944/preprints202305.0311.v1

Abstract

Formalized Calculation, using auxiliary form to calculate various nested sums, based on binary or Gaussian coefficients and providing three different expressions. In addition to computation, it introduces a large number of new concepts and methods for analysis. With just a little skill, it can easily obtain dozens of theorems. It has undergone three years of development. This article summarizes it and provides a large number of latest achievements, showcasing many analytical methods. This article includes various relationships between two types of Stirling numbers, associated Stirling numbers, generalized Euler numbers and polynomials, and provides a theorem of symmetry and extended Wolstenholme theorems.

Keywords

Formal Calculation; nested sums; Gaussian coefficient; Stirling number; associated Stirling numbers; Eulerian number and polynomial; Wolstenholme theorem

Subject

Computer Science and Mathematics, Discrete Mathematics and Combinatorics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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