On the link between binomial theorem and discrete convolution

TL;DR

This paper introduces a new polynomial related to odd powers, establishing identities that connect the binomial theorem with discrete convolution, and extends these results to multinomial cases, verified through computational tools.

Contribution

The paper presents a novel polynomial P_b^m(x), derives identities linking binomial theorem and discrete convolution of odd powers, and extends the framework to multinomials.

Findings

Established polynomial identities for odd powers.

Connected binomial theorem with discrete convolution.

Extended results to multinomial cases.

Abstract

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.