Integral zeroes of Krawtchouk polynomials

Abstract

Krawtchouk polynomials appear in many various areas of mathematics starting from discrete mathematics (e.g., in coding theory), association schemes, and in the theory of graph representations. The existence/non-existence of integral zeroes of these polynomials is crucial for the existence/non-existence of combinatorial structures in the Hamming association schemes. The integer zeroes of Krawtchouk polynomials for k = 4; 5; 6 and 7 have been found using some very recent results on solvability of polynomial diophantine equations. Our aim is two-fold: Firstly, to verify these results using extensive computer calculations. This requires the solution of some of Pell’s equations and the use of the symbolic mathematics software mathematica. Secondly, we numerically investigate a conjecture dealing with the integer zeroes of the Krawtchouk polynomials Pm2 (m^2) (x) and provide confirmation of the conjecture using a combination of approaches up to m <= 1000, i.e., for the polynomials up to degree of about half a million.