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⚠️WarningArchived in favour toUnexpected polynomial identities arising from a classical interpolation problem

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 evolutionof 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 derivativesfor odd powers and finding polynomial derivatives via a double limit.Finally, the manuscript proposes future research directions.

Open research opportunities are described in the section "Future research" ofcurrent manuscript,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 identitiesin terms of discrete convolution may be found
• All kind of derivatives (forward, backward, central), including time scale ones can beexpressed as double limit similarly to [20]
• Equation (1.8) approximates odd-power$2m+1$ in some neighborhood of fixed pointa 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 toPDF

• Finding the derivative of polynomials via double limit (2024)
• Polynomial identity involving Binomial Theorem and Faulhaber's formula (2024)
• Another approach to get derivative of odd-power (2023)
• On the link between binomial theorem and discrete convolution (2016-2022)
• A study on partial dynamic equation on time scales involving derivatives of polynomials (2016-2022)

• Probably some unusual polynomial interpolation approach?

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