https://github.com/kolosovpetro/onthebinomialtheoremanddiscreteconvolution

On the link between binomial theorem and discrete convolution
https://github.com/kolosovpetro/onthebinomialtheoremanddiscreteconvolution

binomial-theorem math mathematics open-research open-science polynomials research research-project

Last synced: 26 days ago
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On the link between binomial theorem and discrete convolution

• Host: GitHub
• URL: https://github.com/kolosovpetro/onthebinomialtheoremanddiscreteconvolution
• Owner: kolosovpetro
• License: mit
• Created: 2022-01-20T10:04:52.000Z (about 3 years ago)
• Default Branch: master
• Last Pushed: 2024-09-25T12:25:18.000Z (5 months ago)
• Last Synced: 2024-11-15T17:44:33.738Z (3 months ago)
• Topics: binomial-theorem, math, mathematics, open-research, open-science, polynomials, research, research-project
• Language: TeX
• Homepage: https://kolosovpetro.github.io/pdf/OnTheBinomialTheoremAndDiscreteConvolution.pdf
• Size: 4.98 MB
• Stars: 1
• Watchers: 2
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Changelog: CHANGELOG.md
  • License: LICENSE

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README

        
# On the link between binomial theorem and discrete convolution

Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in \mathbb{R}$,

$$

    \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r

$$

where $\mathbf{A}_{m,r}$ are real coefficients.

In this manuscript, we introduce the polynomial $\mathbf{P}^{m}_{b}(x)$ and study its properties,

establishing a polynomial identity for odd-powers in terms of this polynomial.

Based on mentioned polynomial identity for odd-powers,

we explore the connection between the Binomial theorem and discrete convolution of odd-powers,

further extending this relation to the multinomial case.

All findings are verified using Mathematica programs.

## Build and run in Intellij IDEA

- 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

      ![LaTeX Configuration](./img/latex_configuration.PNG "LaTeX Configuration")

    - BibTeX Configuration

      ![BibTeX Configuration](./img/bibtex_configuration.PNG "BibTeX Configuration")

- Configure Inverse Search in `Intellij IDEA` for SumatraPDF: `Tools -> LaTeX -> Configure Inverse Search`

- Compile document using `Shift + F10`