On integral zeros of Krawtchouk polynomials
type: Item
instance of: scholarly article
title: On integral zeros of Krawtchouk polynomials
zbMATH Open document ID: 0853.33008
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-07-09T00:00:00Z
review text: The (binary) Krawtchouk polynomials \(P^n_k (x)\) are defined by \(\sum_{k = 0}^\infty P^k_n (x) z^k = (1 - z)^x (1 + z)^{n - x}\). The existence, respectively nonexistence, of integer roots of \(P^n_k (x)\) is of relevance in a number of different problems, like graph reconstruction problems or existence problems for codes. In this paper, for the first time an attempt is made to systematically study the existence of integral roots of Krawtchouk polynomials. The authors provide upper bounds for the number of zeros of Krawtchouk polynomials, give several necessary conditions such that \(P^k_n (x)\) has at least one integer root, and finally present conditions for the nonexistence of integer roots.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 05C60
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 883802
zbMATH Keywords: binary Krawtchouk polynomials
zbMATH Keywords: integral zeros
author: Ilia Krasikov
author: Simon Litsyn
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
full work available at URL: jcta.1996.0038
OpenAlex ID: W2030578446
DOI: 10.1006/JCTA.1996.0038
description: scientific article; zbMATH DE number 883802