On Krawtchouk polynomials

Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. Although the basic properties of these polynomials are known to some extent, there is, to my knowledge, no detailed development available. My aim in writing this article is to fill in this gap. Notation In the following we will use capital letters for (algebraic) polynomials, for example P or P (X); for the polynomial function associated with a polynomial P , we will use small letters in the parentheses, for example P (x). 1 Definition and first properties To begin with, we define a collection of real polynomials P j , for j ≥ 0, which we will use in the definition of Krawtchouk polynomials. We set P 0 = 1 and, for j ≥ 1,