On the link between binomial theorem and discrete convolution

2016/03/08 by Petro Kolosov, Kolosov, Petro
Mathematics · #11C08 #44A35 #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #advanced mathematical theories

paper · doi:10.48550/arxiv.1603.02468

openalex publication_date 2016/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P^m_b(x) be a 2m+1-degree polynomial in x and b ∈ ℝ P^m_b(x) = ∑_k=0^b-1 ∑_r=0^m A_m,r k^r (x-k)^r where A_m,r are real coefficients. In this manuscript, we introduce the polynomial 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.

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