On the link between binomial theorem and discrete convolution

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.

Theorems & Definitions (12)

• Lemma 4.1
• Corollary 4.2
• Lemma 5.1
• Corollary 5.2
• Theorem 5.3
• Proposition 5.4
• Theorem 5.5
• Corollary 5.6
• Corollary 6.1
• Corollary 6.2