  Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree integer-valued polynomial in
$b,x\in\mathbb{R}$ \[
  \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r}
k^r(x-k)^r, \] where $\mathbf{A}_{m,r}$ is a real coefficient. In this
manuscript we establish a relation between Binomial theorem and polynomial
$\mathbf{P}^{m}_{b}(x)$. Furthermore, a relationship between Binomial theorem
and discrete convolution in terms of polynomials is provided.
