On the link between binomial theorem and discrete convolution

Let ${\mathbf{P}}_b^m(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in $\mathbb{R}$$ $${\mathbf{P}}_b^m(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 ${\mathbf{P}}_b^m(x)$ 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.