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The polynomial $\mathbf{P}^m_b(x)$ is a $2m+1$ degree polynomial in $(x,b) \in \mathbb{R}$ defined by an identity for odd-powers. The odd-power identity is derived applying certain interpolation approaches including systems of linear equations and recurrence relations. This manuscript provides a comprehensive historical survey of the milestones and evolution of the polynomial $\mathbf{P}^m_b(x)$ continuing with related works based on it. Notable results inside related works include the relation between ordinary and partial derivatives for odd-powers, finding the derivative of polynomials via double limit etc. Finally, the manuscript concludes with future research directions and activities.

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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