Generalizing Krawtchouk polynomials using Hadamard matrices

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.


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][3][4]; 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 ∈ {0, 1, . . . , }. 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 {0, 1 , 2 , . . . , −1 }, 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 ofpolynomials ( , ); see Definition 2 below. Thesepolynomials 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 orthogonality conditions. In Section 4, we establish recurrence relations for -polynomials. Afterwards, we obtain coefficients for the expansion of a polynomial in terms ofpolynomials in Section 5. Finally, in Section 6, we implement the results obtained here in Mathematica, so the reader may easily derive and explore -polynomials for any matrix .
We use the ℓ 1 -norm (the "taxicab-metric") to measure the length | | of ∈ N 0 , that is, | | = ∑ −1 =0 . We define the set of weak compositions of into numbers by ( , ) = { ∈ N 0 : | | = }; in other words, ( , ) is the subset of -dimensional nonnegative vectors of length . We note that the set ( , ) has cardinality ( (where the convention 0 0 = 1 is used). We note that the multinomial theorem reads In the following, denotes an arbitrary × matrix. We use the following convention to refer to the entries of a × matrix [11]. The entry in the th row and th column is called the ( − 1, − 1) entry, where , = 1, 2, . . . , . Thus, the ( , ) entry of is denoted by , where , = 0, 1, . . . , − 1. Given a matrix , the matrix that remains when the first row and the first column of are removed is called the core of . The next definition is well known. where is the complex conjugate transpose of and is the × identity matrix.

The Generator Polynomial and Orthogonality Relations
The values of the -polynomial with respect to a matrix can be derived from a generator polynomial.
Proof. This is an application of the multinomial theorem (recall that for ( ) ∈ N 0 , we have ( ) = ( ,0 , ,1 , . . . , , −1 )): As an immediate consequence, we can recover the entries of the matrix as the -polynomials of minimum order.

ISRN Applied Mathematics 3
Proof. For = , the left-hand side of (7) becomes and since ∈ (1, ) and thus runs through 0 , . . . , −1 in this case, the rest follows by comparing coefficients of .
Remark 5. Theorem 3 can also be used for summation results of -polynomials over ( , ): using = (1, . . . , 1), we obtain that is, the product of the th power of the th column sum of .
As an immediate result of this remark, we can establish the following corollary.
For a generalized Hadamard matrix , the multinomial theorem yields the following orthogonality relation for the corresponding -polynomials.
Then, on the one hand, we have since is a generalized Hadamard matrix. Consequently, On the other hand, the multinomial theorem yields Equating coefficients of in the two above expressions for , we obtain the desired result.
If is a generalized Hadamard matrix and also satisfies certain additional conditions, then it is possible to establish that the corresponding -polynomials satisfy additional orthogonality conditions. We use the following three results in proving this.

Recurrence Relations and Summation of Certain Polynomials
We use Theorem 3 to derive a recurrence relation forpolynomials in having parameter where both , ∈ ( , ) by a sum of -polynomials where both , ∈ ( − 1, ).
where the sum on the right is taken over all such that the th component of ∈ ( , ) is positive.
Proof. Using (7) twice, we obtain (if > 0) Multiplying the right-hand side out and equating coefficients of establish the claim.
where the sum on the right is taken over all such that the th component of ∈ ( , ) is positive.
Proof. We note that we have Thus, on the one hand, the derivative of the left-hand side of (7) with respect to is ] . (25) Using (7), this is equal to On the other hand, the derivative of the right-hand side of (7) with respect to is by comparing coefficients, we must havẽ= − , and therefore (noting that > 0) We note that (21) and (23) respectively. In particular, Theorem 12 can be easily iterated to obtain the following statements.
where the sum on the right is taken over all ∈ ( , ) such that − is in ( − , ).
Proof. This follows by repeatedly applying (21) in Theorem 12.
The multinomial coefficient ( ) arises as the number of ways can be changed to − step-by-step.
We note that taking = in the previous corollary yields the statement of Definition 2 (since, by definition, (0; 0) = 1).
Combining the recurrence relation in Theorem 12 with Theorem 7 yields the following summation formula forpolynomials.

Expansion of a Polynomial
The coefficients for the expansion of a polynomial in terms of Krawtchouk polynomials have been obtained (cf. [6]). We obtain a similar result for -polynomials defined in terms of a generalized Hadamard matrix, using orthogonality properties.
(i) If is symmetric and the expansion of a polynomial ( ) in terms of -polynomials defined with respect to is (with coefficients ∈ C) then, for all ℓ ∈ ( , ), Similarly, if the expansion of a polynomial ( ) in terms of -polynomials is (with coefficients ∈ C)

Mathematica Code
Here, we present Mathematica code to obtain -polynomials. First, we specify the matrix and the length : From this we can calculate , the set ( , ) and initialize the variable : Thus, we are able to evaluate the -polynomial ( ; ) for any , ∈ ( , ) (and thus check orthogonality and recurrence relations). With the help of the Mathematica function Fit, we can express the m-polynomials as univariant polynomials in ( 0 − 1 ) as follows (we also recall that |( 0 , 1 )| = 0 + 1 = 6 here): ) .
The above mathematica code using Theorem 17 has to be modified as follows: >

Outlook
Krawtchouk polynomials and their generalisation appear in many areas of Mathematics, see [5]: harmonic analysis [1,2,4], statistics [3], combinatorics and coding theory 8 ISRN Applied Mathematics [6,7,9,11,14,15], probability theory [5], representation theory (e.g., of quantum groups) [10,12,13], difference equations [16], and pattern recognition [17] (a website called the "Krawtchouk Polynomials Home Page" by V. Zelenkov at http://orthpol.narod.ru/eng/index.html collecting material about M. Krawtchouk and the polynomials that bear his name has, unfortunately, not been updated in a while). However, our motivation in this paper was primarily driven by generalising the results obtained in [8]-and thus shedding more light on the qualitative structure of (generalisations of) Krawtchouk polynomials-and not yet with a specific application in mind. By providing the Mathematica code to obtain -polynomials, we also hope that other researchers are encouraged to explore them and see if they can be used in their research.