A Note on Krawtchouk Polynomials and Riordan Arrays

The Krawtchouk polynomials play an important role in various areas of mathematics. Notable applications occur in coding theory [11], association schemes [4], and in the theory of group representations [21, 22]. In this note, we explore links between the Krawtchouk polynomials and Riordan arrays, of both ordinary and exponential type, and we study integer sequences defined by evaluating the Krawtchouk polynomials at different values of their parameters. The link between Krawtchouk polynomials and exponential Riordan arrays is implicitly contained in the umbral calculus approach to certain families of orthogonal polynomials. We shall look at these links explicitly in the following. The structure of this note is as follows. In the next section, we shall give a brief introduction to the relevant theory of both ordinary and exponential Riordan arrays. We then define the Krawtchouk polynomials, using exponential Riordan arrays, and look at some general properties of these polynomials from this perspective. We then show that for different values of the parameters used in the definition of the Krawtchouk polynomials, there exist interesting families of (ordinary) Riordan arrays.