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.

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