Formal Calculation

Ji Peng

doi:10.20944/preprints202305.0311.v2

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preprints.org > computer science and mathematics > discrete mathematics and combinatorics > doi: 10.20944/preprints202305.0311.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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.v2 Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v2

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MDPI and ACS Style

Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v2

APA Style

Peng, J. (2023). Formal Calculation. Preprints. https://doi.org/10.20944/preprints202305.0311.v2

Chicago/Turabian Style

Peng, J. 2023 "Formal Calculation" Preprints. https://doi.org/10.20944/preprints202305.0311.v2

Abstract

Formal Calculation introduces a way to calculate various nested sums.It uses an auxiliary Form for calculation and provides results in three forms.Besides computation, it is also a powerful tool for analysis and studies various numbers in a unified way.This article contains many results of two types of Stirling numbers, associated Stirling numbers and Eulerian numbers. Formal Calculation provides a method for obtaining combinatorial identities. Applying it to the Gaussian coefficient is a new way for the analysis of q - Binomial. In the process of analysis, this paper makes a great generalization of Euler numbers and polynomials, Wilson's theorem and Wolstenholme's theorem, revealing that they are just special cases. Finally, this article introduces a theorem on symmetry.

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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Comments (1)

Comment 1

Received: 2 August 2023
Commenter: Ji Peng
Commenter's Conflict of Interests: Author

Comment: Changed to Latex format with some text polishing.

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