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https://github.com/kolosovpetro/historyandoverviewofpolynomialp

Open research project on polynomial interpolation and approximation
https://github.com/kolosovpetro/historyandoverviewofpolynomialp

approximation approximation-algorithms binomial-theorem differential-equations finite-difference interpolation-methods math mathematics maths open-research open-science polynomial-equations polynomials research research-project

Last synced: about 2 months ago
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Open research project on polynomial interpolation and approximation

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README 

        
## History and overview of the polynomial $\mathbf{P}^m_b(x)$



### Abstract



The polynomial $\mathbf{P}^m_b(x)$ is a polynomial of degree $2m+1$ in $(x,b) \in \mathbb{R}$,

defined by an identity for odd powers, closely linked to Binomial theorem and Faulhaber's formula.

The odd-power identity is derived using certain interpolation techniques,

including systems of linear equations, recurrence relations, and finite differences.

This manuscript offers a comprehensive historical survey of the milestones and evolution

of the polynomial $\mathbf{P}^m_b(x)$, followed by related works based on it.

Notable results in related works include the relation between ordinary and partial derivatives

for odd powers and finding polynomial derivatives via a double limit.

Finally, the manuscript proposes future research directions.



### Open research opportunities & other activities



Open research opportunities are described in the section "Future research" of

[current manuscript](https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf),

still it is worth to duplicate them here



- Differential equation (2.1) can be expressed in terms of backward and central differentials,

  as well as its dynamic equation analogs [19]

- Definition (1.8) is closely related to discrete convolution, probably some new identities

  in terms of discrete convolution may be found

- All kind of derivatives (forward, backward, central), including time scale ones can be

  expressed as double limit similarly to [20]

- Equation (1.8) approximates odd-power $2m+1$ in some neighborhood of fixed point

  a as it shown on graphs

- Validation and fixing grammar issues



Please note that equation and references numbers in README may not match the actual in manuscript,

so refer directly to [PDF](https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf)



### Related projects



- [Finding the derivative of polynomials via double limit (2024)](https://github.com/kolosovpetro/FindingTheDerivativeOfPolynomialsViaDoubleLimit)

- [Polynomial identity involving Binomial Theorem and Faulhaber's formula (2024)](https://github.com/kolosovpetro/PolynomialIdentityInvolvingBTandFaulhaber)

- [Another approach to get derivative of odd-power (2023)](https://github.com/kolosovpetro/AnotherApproachToGetDerivativeOfOddPower)

- [On the link between binomial theorem and discrete convolution (2016-2022)](https://github.com/kolosovpetro/OnTheBinomialTheoremAndDiscreteConvolution)

- [A study on partial dynamic equation on time scales involving derivatives of polynomials (2016-2022)](https://github.com/kolosovpetro/AStudyOnDynamicEquations)



### Reddit topic



- [Probably some unusual polynomial interpolation approach?](https://www.reddit.com/r/math/comments/1afo1y3/probably_some_unusual_polynomial_interpolation/)



### ⭐ Support open science on GitHub ⭐



If you found this repository helpful or interesting, please consider giving it a star! ⭐



This simple gesture goes a long way in supporting the project and acknowledging the effort put into its development.



Your star helps boost the project's visibility, making it easier for others to discover and benefit from it as well.

Plus, it serves as motivation for the maintainers to continue improving and adding new features.



### Contributing



Contributions are welcome! 🎉



Please refer to open issues or propose your own changes.



It's better to [create an issue](https://github.com/kolosovpetro/HistoryAndOverviewOfPolynomialP/issues/new)

with description of your potential contribution before creating pull requests.



### Build the manuscript



Refer to the documentation: [BUILD.md](BUILD.md)



### Licensing



This project is licensed under the GPL-3.0 License - see the [LICENSE](LICENSE) file for details.