Abstract

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 [2–4]; 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 . 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 orthogonality conditions. In Section 4, we establish recurrence relations for -polynomials. Afterwards, we obtain coefficients for the expansion of a polynomial in terms of -polynomials 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 .

2. Definitions and Notations

In this paper, denotes . We use the convention that if , then has components . So, if then denotes . We will also use the   elementary unit vectors  , .

We use the -norm (the “taxicab-metric”) to measure the length   of , that is, . We define the set of weak compositions of into numbers by ; in other words, is the subset of -dimensional nonnegative vectors of length . We note that the set has cardinality . For , we use the multi-index notation . Similarly, for a variable and , we write (where the convention 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   entry, where . Thus, the entry of is denoted by , where . 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.

Definition 1. For an integer greater than one, a matrix is called a generalized Hadamard matrix if , where is the complex conjugate transpose of and is the identity matrix.
We now define the -polynomial with respect to a matrix .

Definition 2. Let and be elements of . The  -polynomial in with respect to having parameter is denoted by   or and given by where the summation is taken over all sets of nonnegative integers (with ) satisfying

3. The Generator Polynomial and Orthogonality Relations

The values of the -polynomial with respect to a matrix can be derived from a generator polynomial.

Theorem 3 (generator polynomial). Let and let be a matrix. Then, where .

Proof. This is an application of the multinomial theorem (recall that for , we have ): where .

As an immediate consequence, we can recover the entries of the matrix as the -polynomials of minimum order.

Corollary 4. We have .

Proof. For , the left-hand side of (7) becomes and since and thus runs through 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 , 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.

Corollary 6. If for one (i.e., one of the row-sums of the matrix is zero), then .

For a generalized Hadamard matrix , the multinomial theorem yields the following orthogonality relation for the corresponding -polynomials.

Theorem 7 (orthogonality relation). If is a generalized Hadamard matrix, then the -polynomials , with , satisfy the orthogonality relations where denotes Kronecker’s delta.

Proof. Let , , and define 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.

Lemma 8. If is symmetric, that is, , then .

Proof. By Definition 2, we have (where and ) where we define (and thus and ), and since is symmetric (i.e., ).

The following results can be obtained in a similar manner to Lemma 8.

Lemma 9. If   has a symmetric core, for and for , then .

Such symmetry relations yield additional orthogonality relations.

Theorem 10 (additional orthogonality relation). Let be a generalized Hadamard matrix. (i)If in addition is symmetric, then (ii)If in addition has a symmetric core, for and for , then

Proof. This follows immediately from Theorem 7 and Lemmas 8 and 9.

Remark 11. It turns out that a slight modification of the above -polynomials has already been considered in [12, 13]. For a matrix and , [13, Equation (2.3)] defines polynomials by in our notation. For these polynomials, the following multiplication property is proved (see [13, Theorem 3.2]): for matrices , . With , and thus , Theorem 7 becomes a special case of this equation by noting that (see [13, Equation (4.5)]) and (using the notation of Definition 2, the polynomial on the left can only be nonzero if whenever and otherwise).

4. Recurrence Relations and Summation of Certain Polynomials

We use Theorem 3 to derive a recurrence relation for -polynomials in having parameter where both by a sum of -polynomials where both .

Theorem 12 (recurrence relations I). Let with for one . Then, 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 ) Multiplying the right-hand side out and equating coefficients of establish the claim.

Theorem 13 (recurrence relations II). Let with for one . Then, 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 Using (7), this is equal to On the other hand, the derivative of the right-hand side of (7) with respect to is Equating the last two expressions and interchanging the order of summation yield by comparing coefficients, we must have , and therefore (noting that )

We note that (21) and (23) can be rewritten as respectively.

In particular, Theorem 12 can be easily iterated to obtain the following statements.

Proposition 14. Let with for one . Then, 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.

Corollary 15. Let with for all . Then, where the sum on the right is taken over all the vectors such that .