Polynomial identities involving Central Factorial numbers

Central factorial numbers appear in mathematical literature quite frequently. Few of prominent works are Riordan's Combinatorial identities and Knuth's Johann Faulhaber and sums of powers, along as many others. In this manuscript, we start our discussion from the definition of central factorial numbers (both, recursive and iterative) continuing with a set of identities shown further in this manuscript. Afterward, based on odd-power identities given by Knuth, we show new variations of odd-power identities applying certain relations in terms of central factorial numbers. Finally, we provide a comprehensive way to validate the results of the manuscript via supplementary Mathematica programs.

GitHub Actions used

• https://github.com/actions/checkout
• https://github.com/GitTools/actions/blob/main/docs/examples/github/gitversion/index.md
• https://github.com/xu-cheng/latex-action
• https://github.com/actions/upload-artifact

Build and run using PowerShell (Windows)

• Install MikTeX: https://miktex.org/download
• Update MikTeX
• Install SumatraPDF viewer: https://www.sumatrapdfreader.org/download-free-pdf-viewer
• Rename-Tex-Files.ps1 renames main LaTeX and BibTeX files to match the repository root directory name
• Build-Latex.ps1

Build and run in Intellij IDEA (Windows)

• Install MikTeX: https://miktex.org/download
• Update MikTeX
• Install SumatraPDF viewer: https://www.sumatrapdfreader.org/download-free-pdf-viewer
• Path to SumatraPDF: C:\Program Files\SumatraPDF
• Install Intellij IDEA Ultimate: https://www.jetbrains.com/idea/download/#section=windows
• Activate Intellij IDEA Ultimate
• Install TeXiFy IDEA plugin: https://plugins.jetbrains.com/plugin/9473-texify-idea
• Clone this repository locally: https://github.com/kolosovpetro/github-latex-template.git
• Open github-latex-template folder in Intellij IDEA Ultimate and configure as follows
  • LaTeX Configuration
  • BibTeX Configuration
• Configure Inverse Search in Intellij IDEA for SumatraPDF: Tools -> LaTeX -> Configure Inverse Search
• Compile document using Shift + F10

How to use Mathematica package

• Open the package file GithubLatexTemplateMathematicaPackage.m in Wolfram Mathematica, I use version 13.0
• Execute the package using Shift+Enter
• Open the notebook file GithubLatexTemplateMathematicaNotebook.nb
• Execute the line: Needs["GithubLatexTemplateMathematicaPackage"]
• Continue your work as desired

Configure CI / CD

• Update root file name in build-pdf.yml and build-and-deploy-pdf.yml
• Set repository secrets
  • GH_ACCESS_TOKEN: Generate GitHub Personal access token at Settings -> Developer Settings -> Personal access tokens -> Generate mew token

Actions and their trigger policy

• build-pdf.yml builds project using TeXLive. Triggered on pull_request, push to develop branch
• build-and-deploy-pdf.yml builds project using TeXLive and deploys to GitHub Pages. Triggered on push to main branch