On the link between binomial theorem and discrete convolution

Abstract

Let ${\mathbf{P}}_b^m(x)$ 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 ${\mathbf{P}}_b^m(x)$ 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.

25 pages, source code of the manuscript is available at https://github.com/kolosovpetro/OnTheBinomialTheoremAndDiscreteConvolution