Hindawi Publishing Corporation ISRN Applied Mathematics Volume 2014, Article ID 498135, 8 pages http://dx.doi.org/10.1155/2014/498135 Research Article Generalizing Krawtchouk Polynomials Using Hadamard Matrices Peter S. Chami,1 Bernd Sing,1 and Norris Sookoo2 1 Department of Computer Science, Mathematics and Physics, Faculty of Science and Technology, The University of the West Indies, Cave Hill, St. Michael, Barbados 2 The University of Trinidad and Tobago, O’Meara Campus, Arima, Trinidad and Tobago Correspondence should be addressed to Norris Sookoo; [email protected] Received 15 November 2013; Accepted 22 December 2013; Published 4 March 2014 Academic Editors: F. Ding and X. Liu Copyright © 2014 Peter S. Chami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 where is a natural number and ∈ {0,1,...,}.The generator polynomial is Matrices have been the subject of much study, and large bodies of results have been obtained about them. We study (1+)−(1−) = ∑ () . the interplay between the theory of matrices and the theory (2) of orthogonal polynomials. For Krawtchouk polynomials, =0 introduced in [1], interesting results have been obtained in The generalized Krawtchouk polynomial () is obtained by [2–4]; also see the review article [5]andcompare[6]for generalizing the above generator polynomial as follows: generalized Krawtchouk polynomials. More recently, condi- tions for the existence of integral zeros of binary Krawtchouk −1 −1 polynomials have been obtained in [7], while properties for ∏(∑()) = ∑ () , (3) generalized Krawtchouk polynomials can be found in [8]. =0 =0 ∈(,) Other generalizations of binary Krawtchouk polynomials have also been considered; for example, some properties of where ∑ =∑ =, is a prime power, the are binary Krawtchouk polynomials have been generalised to - indeterminate, the field () with elements is {0, 1, Krawtchouk polynomials in [9]. Orthogonality relations for 2,...,−1},and is a character. quantum and -Krawtchouk polynomials have been derived The above information about Krawtchouk polynomials in [10], and it has been shown that affine -Krawtchouk poly- and generalized Krawtchouk polynomials was taken from [6]. nomials are dual to quantum -Krawtchouk polynomials. In If we replace the () by arbitrary scalars in the this paper, we define and study generalizations of Krawtchouk lastequation,weobtainthegeneratorpolynomialof- polynomials, namely, -polynomials. polynomials (,;see ) Definition 2 below. These - The Krawtchouk polynomial () is given by polynomialsarethesubjectofstudyinthispaper. In Section 2, we present relevant notations and defini- − tions. In Section 3, we introduce the generator polynomial. () = ∑(−1) ( )( ) , − (1) The associated matrix of coefficients can be any square =0 matrix, and so the question that immediately arises is how 2 ISRN Applied Mathematics the properties of the -polynomials are related to the respect to having parameter is denoted by (; ) or properties of .Wewillestablishthat,if is a generalized (0,1,...,−1;0,1,...,−1) and given by Hadamard matrix, then the associated -polynomials satisfy −1 −1 orthogonality conditions. In Section 4,weestablishrecur- ! (; )= ∑ ∏∏ , rence relations for -polynomials.Afterwards,weobtain ∏ ! coefficients for the expansion of a polynomial in terms of - , , , =0 =0 (5) polynomials in Section 5.Finally,inSection 6,weimplement −1 −1 , the results obtained here in Mathematica, so the reader may =!∑∏∏ , ! easily derive and explore -polynomials for any matrix . , =0 =0 , where the summation is taken over all sets of nonnegative ,=0,1,...,−1 2. Definitions and Notations integers , (with )satisfying In this paper, N0 denotes {0,1,2,3,...}.Weusetheconven- −1 tion that if ∈N0,then has components (0,1,...,−1). ∑, =, =0,1,...,−1, =0 So, if () ∈ N then () denotes (,0,,1,...,,−1).Wewill 0 (6) also use the elementary unit vectors 0 = (1, 0, 0, . , 0), −1 =(0,1,0,...,0),..., =(0,0,...,0,1) 1 −1 . ∑, =, =0,1,...,−1. 1 We use the ℓ -norm (the “taxicab-metric”) to measure the =0 −1 length || of ∈N0,thatis,|| = ∑=0 . We define the set of weak compositions of into numbers by (, ) = 3. The Generator Polynomial and { ∈ N :||=} (, ) 0 ;inotherwords, is the subset Orthogonality Relations of -dimensional nonnegative vectors of length .Wenote +−1 that the set (, ) has cardinality ( −1 ).For∈N0,we The values of the -polynomial with respect to a matrix −1 can be derived from a generator polynomial. use the multi-index notation ! = ∏=0 !.Similarly,fora =(, ,..., )∈C ∈N variable 0 1 −1 and 0,wewrite Theorem 3 =(, ,..., )∈ −1 (generator polynomial). Let 0 1 −1 =∏ 00 =1 =0 (where the convention is used). We note (, ) and let =( ) be a ×matrix. Then, that the multinomial theorem reads −1 −1 ∏(∑ ) = ∑ (; ) , (7) =1 =0 ∈(,) −1 (∑ ) = ∑ ( )⋅ =(, ,..., ) =0 ∈(,) where 0 1 −1 . (4) ! −1 Proof. This is an application of the multinomial theorem 0 = ∑ ⋅0 ⋅⋅⋅−1 . !⋅⋅⋅ ! (recall that for () ∈ N0,wehave() =(,0,,1,...,,−1)): ∈(,) 0 −1 −1 −1 −1 −1 { ! } [ , ] ∏(∑ ) = ∏ { ∑ ∏( ) } =0 =0 =0 ()! =0 In the following, denotes an arbitrary ×matrix. We {()∈(,) [ ]} use the following convention to refer to the entries of a × −1 −1 ! matrix [11]. The entry in the th row and th column is called = ∑ (∏∏ , ) the (−1,−1)entry,where, = 1,2,...,.Thus,the(, ) ∏ ! ∈ , , , , =1 =0 ,=0,1,...,−1 () ( ) entry of is denoted by ,where .Given =0,1,...,−1 (8) amatrix, the matrix that remains when the first row and thefirstcolumnof are removed is called the core of . −1 0,+1,+⋅⋅⋅+−1, The next definition is well known. × ∏ =0 × 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. where =0, +1, +⋅⋅⋅+−1,. We now define the -polynomial with respect to a matrix . As an immediate consequence, we can recover the entries of the matrix as the -polynomials of minimum order. Definition 2. Let =(0,1,,...,−1) and =(0,1,..., Corollary 4. ( ; )= −1) be elements of (,. ) The -polynomial in with We have ℓ ℓ. ISRN Applied Mathematics 3 Proof. For =, the left-hand side of (7)becomes On the other hand, the multinomial theorem yields + +⋅⋅⋅+ , ,0 0 ,1 1 ,−1 −1 (9) ! −1 −1 −1 ∈(1,) ,..., = ∑ ∏(∑ ) (∑ ) and since and thus runs through 0 −1 in ! this case, the rest follows by comparing coefficients of . ∈(,) =0 =0 =0 Remark 5. Theorem 3 canalsobeusedforsummationresults ! (, ) =(1,...,1) = ∑ [ ∑ (; )] of -polynomials over :using ,weobtain ! (16) −1 −1 ∈(,) [∈(,) ] ∑ (; )= ∏(∑ ) , (10) ∈(,) =0 =0 × [ ∑ (;) ] . that is, the product of the th power of the th column sum [∈(,) ] of . As an immediate result of this remark, we can establish Equating coefficients of in the two above expressions the following corollary. for , we obtain the desired result. Corollary 6. ∑−1 =0 0≤≤−1 If =0 for one If is a generalized Hadamard matrix and also satisfies (i.e., one of the row-sums of the matrix is zero), then certain additional conditions, then it is possible to establish ∑∈(,) (; ). =0 that the corresponding -polynomials satisfy additional orthogonality conditions. We use the following three results For a generalized Hadamard matrix ,themultinomial in proving this. theorem yields the following orthogonality relation for the corresponding -polynomials. Lemma 8. If is symmetric, that is, = ,then(; )= (!/!)(; ). Theorem 7 (orthogonality relation). If is a generalized Hadamard matrix, then the -polynomials (, ),with −1 Proof. By Definition,wehave(where 2 ∑=0 ℎ, = and , ∈ (,, ) satisfy the orthogonality relations −1 ∑ ℎ, =) 1 =0 ∑ (; ) (;) = ,, ! ! (11) −1 −1 ℎ, ∈(,) (; ) =! ∑∏∏ ℎ,! where ℎ, =0 =0 =(0,1,...,−1),=(0,1,...,−1), ℎ ! −1 −1 , = !∑∏∏ =(, ,..., ), ! ℎ ! 0 1 −1 ℎ =0 =0 , (12) , (17) −1 −1 −1 1, = ∀, ! , , = ∏ , ={ 0, ℎ, = !∑∏∏ =0 ! ! , =0 =0 , denotes Kronecker’s delta. ! = (; ), Proof. Let =(0,1,...,−1), =(0,1,...,−1)∈C , ! and define −1 −1 −1 −1 −1 −1 −1 where we define , =ℎ, (and thus ∑=0 , = and =∑ (∑ )(∑ )=∑ ∑ ∑ . −1 ∑=0 , =), and since is symmetric (i.e., =). =0 =0 =0 =0 =0 =0 (13) The following results can be obtained in a similar manner Then, on the one hand, we have to Lemma 8. −1 Lemma 9. If has a symmetric core, 0, =1for = =∑ (14) 0,...,−1and ,0 =−1for =1,...,−1,then(; )= =0 + (−1) 0 0 (!/!)(; ). since is a generalized Hadamard matrix. Consequently, Such symmetry relations yield additional orthogonality { ! } = ∑ ( ) 0 ( ) 1 ⋅⋅⋅( ) −1 . relations. { ! 0 0 1 1 −1 −1 } ∈(,) { } Theorem 10 (additional orthogonality relation). Let be a (15) generalized Hadamard matrix. 4 ISRN Applied Mathematics (i) If in addition is symmetric, then Proof. Using (7)twice,weobtain(if >0) ∑ (; ) ∑ (; ) (;) = . ∈(,) , (18) ∈ , ( ) −1 −1 = ∏(∑ ) =1 =0 (ii) If in