%0 Journal Article
%T Generalizing Krawtchouk Polynomials Using Hadamard Matrices
%A Peter S. Chami
%A Bernd Sing
%A Norris Sookoo
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/498135
%X We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica. 1. Introduction Matrices have been the subject of much study, and large bodies of results have been obtained about them. We study the interplay between the theory of matrices and the theory of orthogonal polynomials. For Krawtchouk polynomials, introduced in [1], interesting results have been obtained in [2¨C4]; also see the review article [5] and compare [6] for generalized Krawtchouk polynomials. More recently, conditions for the existence of integral zeros of binary Krawtchouk polynomials have been obtained in [7], while properties for generalized Krawtchouk polynomials can be found in [8]. Other generalizations of binary Krawtchouk polynomials have also been considered; for example, some properties of binary Krawtchouk polynomials have been generalised to -Krawtchouk polynomials in [9]. Orthogonality relations for quantum and -Krawtchouk polynomials have been derived in [10], and it has been shown that affine -Krawtchouk polynomials are dual to quantum -Krawtchouk polynomials. In this paper, we define and study generalizations of Krawtchouk polynomials, namely, -polynomials. The Krawtchouk polynomial is given by where is a natural number and . The generator polynomial is The generalized Krawtchouk polynomial is obtained by generalizing the above generator polynomial as follows: where , is a prime power, the are indeterminate, the field with elements is , and is a character. The above information about Krawtchouk polynomials and generalized Krawtchouk polynomials was taken from [6]. If we replace the by arbitrary scalars in the last equation, we obtain the generator polynomial of -polynomials ; see Definition 2 below. These -polynomials are the subject of study in this paper. In Section 2, we present relevant notations and definitions. In Section 3, we introduce the generator polynomial. The associated matrix of coefficients can be any square matrix, and so the question that immediately arises is how the properties of the -polynomials are related to the properties of . We will establish that, if is a generalized Hadamard matrix, then the associated -polynomials satisfy
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/498135/