Generalizing Krawtchouk polynomials using Hadamard matrices

TL;DR

This paper introduces a new class of polynomials called m-polynomials, explores their orthogonality properties, and provides computational implementations, expanding the theoretical framework of Krawtchouk polynomials using Hadamard matrices.

Contribution

It generalizes Krawtchouk polynomials through the concept of m-polynomials with matrix-structured coefficients and establishes their orthogonality relations.

Findings

Derived orthogonality relations for m-polynomials.

Provided explicit coefficients for polynomial expansions.

Implemented m-polynomials in MATHEMATICA and demonstrated results.

Abstract

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.