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 are described in the section "Future research" of current 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 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

• 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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