Formal Calculation

Ji Peng

doi:10.20944/preprints202305.0311.v3

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preprints.org > computer science and mathematics > discrete mathematics and combinatorics > doi: 10.20944/preprints202305.0311.v3

Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed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.v3 Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v3

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MDPI and ACS Style

Peng, J. Formal Calculation. Preprints 2023, 2023050311. https://doi.org/10.20944/preprints202305.0311.v3

APA Style

Peng, J. (2023). Formal Calculation. Preprints. https://doi.org/10.20944/preprints202305.0311.v3

Chicago/Turabian Style

Peng, J. 2023 "Formal Calculation" Preprints. https://doi.org/10.20944/preprints202305.0311.v3

Abstract

Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. In addition to computation, it is also a powerful tool for analysis, allowing one to study various numbers in a unified way. This article contains many results of two types of Stirling numbers, associated Stirling numbers, and Eulerian numbers, making a great generalization of Euler polynomials, Wilson's theorem, and Wolstenholme's theorem, showing that they are just special cases. Formal Calculation provides a novel method for obtaining combinatorial identities and analyzing q-binomial.This article has obtained a large number of results in q-analogues, including inversion formulas for q-binomial coefficients. This article also 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: 7 October 2023
Commenter: Ji Peng
Commenter's Conflict of Interests: Author

Comment: Add a large number of results in q-analogue, including inversion formulas of q-binomial

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