Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Formal Calculation

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

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

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.
+ Respond to this comment

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 1


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.