On integral zeros of Krawtchouk polynomials
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.
scientific article; zbMATH DE number 883802