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 𝑛 is a natural number and 𝑥 ∈ {0,1,...,𝑛}.The generator polynomial is Matrices have been the subject of much study, and large bodies of results have been obtained about them. We study 𝑛 (1+𝑧)𝑛−𝑥(1−𝑧)𝑥 = ∑𝐾 (𝑥) 𝑧𝑘. the interplay between the theory of matrices and the theory 𝑘 (2) of orthogonal polynomials. For Krawtchouk polynomials, 𝑘=0 introduced in [1], interesting results have been obtained in The generalized Krawtchouk polynomial 𝐾𝑝(𝑠) 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- 𝑠 tions for the existence of integral zeros of binary Krawtchouk 𝑞−1 𝑞−1 𝑖 𝑝 polynomials have been obtained in [7], while properties for ∏(∑𝜒(𝜔𝑖𝜔𝑙)𝑧𝑖) = ∑ 𝐾𝑝 (𝑠) 𝑧 , (3) generalized Krawtchouk polynomials can be found in [8]. 𝑙=0 𝑖=0 𝑝∈𝑉(𝑛,𝑞) Other generalizations of binary Krawtchouk polynomials have also been considered; for example, some properties of where ∑𝑠𝑖 =∑𝑝𝑖 =𝑛, 𝑞 is a prime power, the 𝑧𝑖 are binary Krawtchouk polynomials have been generalised to 𝑞- indeterminate, the field 𝐺𝐹(𝑞) with 𝑞 elements is {0, 𝜔1, Krawtchouk polynomials in [9]. Orthogonality relations for 𝜔2,...,𝜔𝑞−1},and𝜒 is a character. quantum and 𝑞-Krawtchouk polynomials have been derived The above information about Krawtchouk polynomials in [10], and it has been shown that affine 𝑞-Krawtchouk poly- and generalized Krawtchouk polynomials was taken from [6]. nomials are dual to quantum 𝑞-Krawtchouk polynomials. In If we replace the 𝜒(𝜔𝑖𝜔𝑙) by arbitrary scalars in the this paper, we define and study generalizations of Krawtchouk lastequation,weobtainthegeneratorpolynomialof𝑚- polynomials, namely, 𝑚-polynomials. polynomials 𝑀𝐺(𝑝, 𝑠);seeDefinition 2 below. These 𝑚- The Krawtchouk polynomial 𝐾𝑘(𝑥) is given by polynomialsarethesubjectofstudyinthispaper. In Section 2, we present relevant notations and defini- 𝑘 𝑥 𝑛−𝑥 tions. In Section 3, we introduce the generator polynomial. 𝐾 (𝑥) = ∑(−1)𝑗 ( )( ) , 𝑘 𝑗 𝑘−𝑗 (1) The associated matrix of coefficients 𝐺 can be any square 𝑗=0 matrix, and so the question that immediately arises is how 2 ISRN Applied Mathematics the properties of the 𝑚-polynomials are related to the respect to 𝐺 having parameter 𝑝 is denoted by 𝑀𝐺(𝑝; 𝑠) or properties of 𝐺.Wewillestablishthat,if𝐺 is a generalized 𝑀𝐺(𝑝0,𝑝1,...,𝑝𝑞−1;𝑠0,𝑠1,...,𝑠𝑞−1) and given by Hadamard matrix, then the associated 𝑚-polynomials satisfy 𝑞−1 𝑞−1 orthogonality conditions. In Section 4,weestablishrecur- 𝑠! 𝑟 𝑀𝐺 (𝑝; 𝑠)= ∑ ∏∏𝑔 𝑎,𝑏 rence relations for 𝑚-polynomials.Afterwards,weobtain 𝑎𝑏 𝑟 ∏ 𝑟 ! coefficients for the expansion of a polynomial in terms of 𝑚- 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 𝑎=0 𝑏=0 (5) polynomials in Section 5.Finally,inSection 6,weimplement 𝑟 𝑞−1 𝑞−1 𝑔 𝑎,𝑏 the results obtained here in Mathematica, so the reader may =𝑠!∑∏∏ 𝑎𝑏 , 𝑚 𝐺 𝑟 ! easily derive and explore -polynomials for any matrix . 𝑟𝑖,𝑗 𝑎=0 𝑏=0 𝑎,𝑏

where the summation is taken over all sets of nonnegative 𝑟 𝑖,𝑗=0,1,...,𝑞−1 2. Definitions and Notations integers 𝑖,𝑗 (with )satisfying

In this paper, N0 denotes {0,1,2,3,...}.Weusetheconven- 𝑞−1 𝑞 tion that if 𝑝∈N0,then𝑝 has components (𝑝0,𝑝1,...,𝑝𝑞−1). ∑𝑟𝑖,𝑗 =𝑠𝑖, 𝑖=0,1,...,𝑞−1, 𝑞 𝑗=0 So, if 𝑟(𝑖) ∈ N then 𝑟(𝑖) denotes (𝑟𝑖,0,𝑟𝑖,1,...,𝑟𝑖,𝑞−1).Wewill 0 (6) also use the 𝑞 elementary unit vectors 𝑒0 = (1, 0, 0, . . . , 0), 𝑞−1 𝑒 =(0,1,0,...,0),...,𝑒 =(0,0,...,0,1) 1 𝑞−1 . ∑𝑟𝑖,𝑗 =𝑝𝑗, 𝑗=0,1,...,𝑞−1. 1 We use the ℓ -norm (the “taxicab-metric”) to measure the 𝑖=0 𝑞 𝑞−1 length |𝑝| of 𝑝∈N0,thatis,|𝑝| = ∑𝑖=0 𝑝𝑖. We define the set of weak compositions of 𝑛 into 𝑞 numbers by 𝑉(𝑛, 𝑞) = 3. The Generator Polynomial and {𝑝 ∈ N𝑞 :|𝑝|=𝑛} 𝑉(𝑛, 𝑞) 0 ;inotherwords, is the subset Orthogonality Relations of 𝑞-dimensional nonnegative vectors of length 𝑛.Wenote 𝑛+𝑞−1 𝑞 that the set 𝑉(𝑛, 𝑞) has cardinality ( 𝑞−1 ).For𝑝∈N0,we The values of the 𝑚-polynomial with respect to a matrix 𝐺 𝑞−1 can be derived from a generator polynomial. use the multi-index notation 𝑝! = ∏𝑖=0 𝑝𝑖!.Similarly,fora 𝑧=(𝑧,𝑧 ,...,𝑧 )∈C𝑞 𝑝∈N𝑞 variable 0 1 𝑞−1 and 0,wewrite Theorem 3 𝑠=(𝑠,𝑠 ,...,𝑠 )∈ 𝑞−1 𝑝 (generator polynomial). Let 0 1 𝑞−1 𝑧𝑝 =∏ 𝑧 𝑖 00 =1 𝑗=0 𝑗 (where the convention is used). We note 𝑉(𝑛, 𝑞) and let 𝐺=(𝑔𝑖𝑗 ) be a 𝑞×𝑞matrix. Then, that the multinomial theorem reads 𝑠 𝑞−1 𝑞−1 𝑖 𝑝 ∏(∑𝑔𝑖𝑗 𝑧𝑗) = ∑ 𝑀𝐺 (𝑝; 𝑠)𝑧 , (7) 𝑛 𝑖=1 𝑗=0 𝑝∈𝑉(𝑛,𝑞) 𝑞−1 𝑛 (∑𝑧 ) = ∑ ( )⋅𝑧𝑝 𝑖 𝑝 𝑝=(𝑝,𝑝 ,...,𝑝 ) 𝑖=0 𝑝∈𝑉(𝑛,𝑞) where 0 1 𝑞−1 . (4)

𝑛! 𝑝 𝑝𝑞−1 Proof. This is an application of the multinomial theorem 0 𝑞 = ∑ ⋅𝑧0 ⋅⋅⋅𝑧𝑞−1 . 𝑝 !⋅⋅⋅𝑝 ! (recall that for 𝑟(𝑖) ∈ N0,wehave𝑟(𝑖) =(𝑟𝑖,0,𝑟𝑖,1,...,𝑟𝑖,𝑞−1)): 𝑝∈𝑉(𝑛,𝑞) 0 𝑞−1 𝑠 𝑞−1 𝑞−1 𝑖 𝑞−1 𝑞−1 { 𝑠 ! 𝑟 } 𝑖 [ 𝑖,𝑗 ] ∏(∑𝑔𝑖𝑗 𝑧𝑗) = ∏ { ∑ ∏(𝑔𝑖𝑗 𝑧𝑗) } 𝑖=0 𝑗=0 𝑖=0 𝑟(𝑖)! 𝑗=0 In the following, 𝐺 denotes an arbitrary 𝑞×𝑞matrix. We {𝑟(𝑖)∈𝑉(𝑠𝑖,𝑞) [ ]} use the following convention to refer to the entries of a 𝑞×𝑞 𝑞−1 𝑞−1 𝑖 𝑗 𝑠! 𝑟 matrix [11]. The entry in the th row and th column is called = ∑ (∏∏𝑔 𝑎,𝑏 ) the (𝑖−1,𝑗−1)entry,where𝑖,𝑗 = 1,2,...,𝑞.Thus,the(𝑖, 𝑗) ∏ 𝑟 ! 𝑎𝑏 𝑟 ∈𝑉 𝑠 ,𝑞 , 𝑖,𝑗 𝑖,𝑗 𝑎=1 𝑏=0 𝐺 𝑔 𝑖,𝑗=0,1,...,𝑞−1 (𝑖) ( 𝑖 ) entry of is denoted by 𝑖𝑗 ,where .Given 𝑖=0,1,...,𝑞−1 (8) amatrix𝐺, the matrix that remains when the first row and thefirstcolumnof𝐺 are removed is called the core of 𝐺. 𝑞−1 𝑟0,𝑘+𝑟1,𝑘+⋅⋅⋅+𝑟𝑞−1,𝑘 The next definition is well known. × ∏𝑧𝑘 𝑘=0 𝑞 𝑞×𝑞 Definition 1. For an integer greater than one, a matrix 𝑝 𝑇 = ∑ 𝑀𝐺 (𝑝; 𝑠)𝑧 , 𝐻 is called a generalized Hadamard matrix if 𝐻𝐻 =𝑞𝐼𝑞, 𝑝∈𝑉 𝑛,𝑞 𝑇 ( ) where 𝐻 is the complex conjugate transpose of 𝐻 and 𝐼𝑞 is the 𝑞×𝑞identity matrix. where 𝑝𝑘 =𝑟0,𝑘 +𝑟1,𝑘 +⋅⋅⋅+𝑟𝑞−1,𝑘. We now define the 𝑚-polynomial with respect to a matrix 𝐺. As an immediate consequence, we can recover the entries 𝑔𝑖𝑗 of the matrix 𝐺 as the 𝑚-polynomials of minimum order. Definition 2. Let 𝑠=(𝑠0,𝑠1,,...,𝑠𝑞−1) and 𝑝=(𝑝0,𝑝1,..., Corollary 4. 𝑀𝐺(𝑒 ;𝑒 )=𝑔 𝑝𝑞−1) be elements of 𝑉(𝑛,. 𝑞) The 𝑚-polynomial in 𝑠 with We have ℓ 𝑘 𝑘ℓ. ISRN Applied Mathematics 3

Proof. For 𝑠=𝑒𝑘, the left-hand side of (7)becomes On the other hand, the multinomial theorem yields 𝑔 𝑧 +𝑔 𝑧 +⋅⋅⋅+𝑔 𝑧 , 𝑘,0 0 𝑘,1 1 𝑘,𝑞−1 𝑞−1 (9) 𝑠𝑖 𝑠 𝑛! 𝑞−1 𝑞−1 𝑞−1 𝑖 𝑝∈𝑉(1,𝑞) 𝑒 ,...,𝑒 𝐽𝑛 = ∑ ∏(∑𝑔 𝑧 ) (∑𝑔 𝑦 ) and since and thus runs through 0 𝑞−1 in 𝑠! 𝑖𝑗 𝑗 𝑖𝑘 𝑘 this case, the rest follows by comparing coefficients of 𝑧𝑗. 𝑠∈𝑉(𝑛,𝑞) 𝑖=0 𝑗=0 𝑘=0

Remark 5. Theorem 3 canalsobeusedforsummationresults 𝑛! 𝑚 𝑉(𝑛, 𝑞) 𝑧=(1,...,1) = ∑ [ ∑ 𝑀𝐺 (𝑝; 𝑠)𝑧𝑝] of -polynomials over :using ,weobtain 𝑠! (16) 𝑠 𝑞−1 𝑞−1 𝑖 𝑠∈𝑉(𝑛,𝑞) [𝑝∈𝑉(𝑛,𝑞) ] ∑ 𝑀𝐺 (𝑝; 𝑠)= ∏(∑𝑔𝑖𝑗 ) , (10) 𝑝∈𝑉(𝑛,𝑞) 𝑖=0 𝑗=0 × [ ∑ 𝑀𝐺 (𝑡; 𝑠)𝑦𝑡] . that is, the product of the 𝑠𝑖th power of the 𝑖th column sum [𝑡∈𝑉(𝑛,𝑞) ] of 𝐺. 𝑝 𝑡 As an immediate result of this remark, we can establish Equating coefficients of 𝑧 𝑦 in the two above expressions 𝑛 the following corollary. for 𝐽 , we obtain the desired result. Corollary 6. ∑𝑞−1 𝑔 =0 0≤𝑖≤𝑞−1 If 𝑗=0 𝑖𝑗 for one If 𝐺 is a generalized Hadamard matrix and also satisfies (i.e., one of the row-sums of the matrix 𝐺 is zero), then certain additional conditions, then it is possible to establish ∑𝑝∈𝑉(𝑛,𝑞) 𝑀𝐺(𝑝; 𝑠). =0 that the corresponding 𝑚-polynomials satisfy additional orthogonality conditions. We use the following three results For a generalized Hadamard matrix 𝐺,themultinomial in proving this. theorem yields the following orthogonality relation for the 𝑇 corresponding 𝑚-polynomials. Lemma 8. If 𝐺 is symmetric, that is, 𝐺=𝐺 ,then𝑀𝐺(𝑠; 𝑝)= (𝑝!/𝑠!)𝑀𝐺(𝑝; 𝑠). Theorem 7 (orthogonality relation). If 𝐺 is a generalized Hadamard matrix, then the 𝑚-polynomials 𝑀𝐺(𝑝, 𝑠),with 𝑞−1 Proof. By Definition,wehave(where 2 ∑𝑗=0 ℎ𝑖,𝑗 =𝑝𝑖 and 𝑝, 𝑠 ∈ 𝑉(𝑛,, 𝑞) satisfy the orthogonality relations 𝑞−1 ∑ ℎ𝑖,𝑗 =𝑠𝑗) 1 𝑞𝑛 𝑖=0 ∑ 𝑀𝐺 (𝑝; 𝑠) 𝑀𝐺 (𝑡; 𝑠) = 𝛿𝑝,𝑡, 𝑠! 𝑝! (11) 𝑞−1 𝑞−1 ℎ𝑎,𝑏 𝑠∈𝑉(𝑛,𝑞) 𝑔 𝑀𝐺 (𝑠; 𝑝) =𝑝! ∑∏∏ 𝑎𝑏 ℎ𝑎,𝑏! where ℎ𝑖,𝑗 𝑎=0 𝑏=0

𝑠=(𝑠0,𝑠1,...,𝑠𝑞−1),𝑝=(𝑝0,𝑝1,...,𝑝𝑞−1), ℎ 𝑝! 𝑞−1 𝑞−1 𝑔 𝑎,𝑏 = 𝑠!∑∏∏ 𝑏𝑎 𝑡=(𝑡,𝑡 ,...,𝑡 ), 𝑠! ℎ ! 0 1 𝑞−1 ℎ 𝑎=0 𝑏=0 𝑎,𝑏 (12) 𝑖,𝑗 (17) 𝑞−1 𝑞−1 𝑞−1 𝑟 1, 𝑖𝑓𝑖 𝑝 =𝑡𝑖 ∀𝑖, 𝑝! 𝑔 𝑏,𝑎 𝛿𝑝,𝑡 = ∏𝛿𝑝 ,𝑡 ={ 𝑏𝑎 𝑖 𝑖 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, = 𝑠!∑∏∏ 𝑖=0 𝑠! 𝑟 ! 𝑟𝑖,𝑗 𝑏=0 𝑎=0 𝑏,𝑎 denotes Kronecker’s delta. 𝑝! 𝑞 = 𝑀𝐺 (𝑝; 𝑠), Proof. Let 𝑧=(𝑧0,𝑧1,...,𝑧𝑞−1), 𝑦=(𝑦0,𝑦1,...,𝑦𝑞−1)∈C , 𝑠! and define 𝑞−1 𝑞−1 𝑞−1 𝑞−1 𝑞−1 𝑞−1 𝑞−1 where we define 𝑟𝑖,𝑗 =ℎ𝑗,𝑖 (and thus ∑𝑗=0 𝑟𝑖,𝑗 =𝑠𝑖 and 𝐽=∑ (∑𝑔 𝑧 )(∑𝑔 𝑦 )=∑ ∑ ∑𝑔 𝑔 𝑧 𝑦 . 𝑞−1 𝑖𝑗 𝑗 𝑖𝑘 𝑘 𝑖𝑗 𝑖𝑘 𝑗 𝑘 ∑𝑖=0 𝑟𝑖,𝑗 =𝑝𝑗), and since 𝐺 is symmetric (i.e., 𝑔𝑎𝑏 =𝑔𝑏𝑎). 𝑖=0 𝑗=0 𝑘=0 𝑖=0 𝑗=0 𝑘=0 (13) The following results can be obtained in a similar manner Then, on the one hand, we have to Lemma 8. 𝑞−1 Lemma 9. If 𝐺 has a symmetric core, 𝑔0,𝑗 =1for 𝑗= 𝐽=𝑞∑𝑧𝑗𝑦𝑗 (14) 0,...,𝑞−1and 𝑔𝑖,0 =−1for 𝑖=1,...,𝑞−1,then𝑀𝐺(𝑠; 𝑝)= 𝑗=0 𝑠 +𝑝 (−1) 0 0 (𝑝!/𝑠!)𝑀𝐺(𝑝; 𝑠). since 𝐺 is a generalized Hadamard matrix. Consequently, Such symmetry relations yield additional orthogonality { 𝑛! 𝑠 𝑠 𝑠 } 𝐽𝑛 =𝑞𝑛 ∑ (𝑧 𝑦 ) 0 (𝑧 𝑦 ) 1 ⋅⋅⋅(𝑧 𝑦 ) 𝑞−1 . relations. { 𝑠! 0 0 1 1 𝑞−1 𝑞−1 } 𝑠∈𝑉(𝑛,𝑞) { } Theorem 10 (additional orthogonality relation). Let 𝐺 be a (15) generalized Hadamard matrix. 4 ISRN Applied Mathematics

(i) If in addition 𝐺 is symmetric, then Proof. Using (7)twice,weobtain(if𝑠𝑖 >0) ∑ 𝑀𝐺 (𝑝; 𝑠)𝑧𝑝 𝑛 ∑ 𝑀𝐺 (𝑝; 𝑠) 𝑀𝐺 (𝑠;) 𝑡 =𝑞𝛿 . 𝑝∈𝑉(𝑛,𝑞) 𝑝,𝑡 (18) 𝑠∈𝑉 𝑛,𝑞 ( ) 𝑠 𝑞−1 𝑞−1 𝑖

= ∏(∑𝑔𝑖𝑗 𝑧𝑗) 𝑖=1 𝑗=0

𝑠 𝑠 (ii) If in addition 𝐺 has a symmetric core, 𝑔0,𝑗 =1for 𝑗= 𝑞−1 𝑞−1 0 𝑞−1 𝑖−1 0,...,𝑞−1and 𝑔𝑖,0 =−1for 𝑖=1,...,𝑞−1,then =(∑𝑔𝑖𝑗 𝑧𝑗)(∑𝑔0𝑗𝑧𝑗) ⋅⋅⋅(∑𝑔𝑖−1,𝑗𝑧𝑗) , 𝑗=0 𝑗=0 𝑗=0

𝑠0 𝑡0 𝑛 𝑠 −1 𝑠 𝑠 ∑ (−1) 𝑀𝐺 (𝑝; 𝑠) 𝑀𝐺 (𝑠;) 𝑡 = (−1) 𝑞 𝛿 . 𝑞−1 𝑖 𝑞−1 𝑖+1 𝑞−1 𝑞−1 𝑝,𝑡 (19) 𝑠∈𝑉 𝑛,𝑞 ( ) (∑𝑔𝑖𝑗 𝑧𝑗) (∑𝑔𝑖+1,𝑗𝑧𝑗) ⋅⋅⋅(∑𝑔𝑞−1,𝑗𝑧𝑗) 𝑗=0 𝑗=0 𝑗=0

𝑞−1 =(∑𝑔 𝑧 ) ∑ 𝑀𝐺 (𝑝;̃ 𝑠 −𝑒 )𝑧𝑝̃. Proof. This follows immediately from Theorem 7 and Lem- 𝑖𝑗 𝑗 𝑖 𝑗=0 mas 8 and 9. 𝑝∈𝑉̃ (𝑛−1,𝑞) (22) Remark 11. It turns out that a slight modification of the above 𝑚-polynomials has already been considered in [12, 13]. For Multiplying the right-hand side out and equating coefficients 𝑧𝑝 amatrix𝐺 and 𝑝, 𝑠 ∈ 𝑉(𝑛,,[ 𝑞) 13, Equation (2.3)] defines of establish the claim. polynomials 𝐿𝑝,𝑠(𝐺) by 𝐿𝑝,𝑠(𝐺) = (1/𝑠!)𝑀𝐺(𝑝; 𝑠) in our Theorem 13 (recurrence relations II). Let 𝑝, 𝑠 ∈ 𝑉(𝑛, 𝑞) with notation. For these polynomials, the following multiplication 𝑝 >0 𝑘∈{0,...,𝑞−1} property is proved (see [13, Theorem 3.2]): 𝑘 for one .Then, 𝑞−1 𝑠 𝑀𝐺 (𝑝; 𝑠)= ∑ 𝑖 𝑔 𝑀𝐺 (𝑝 −𝑒 ;𝑠−𝑒), 𝑝 𝑖𝑘 𝑘 𝑖 (23) 𝐿𝑝,𝑡 (𝐺1𝐺2)= ∑ 𝑠!𝐿𝑝,𝑠 (𝐺1)𝐿𝑠,𝑡 (𝐺2), 𝑖=0, 𝑘 (20) 𝑠 >0 𝑠∈𝑉(𝑛,𝑞) 𝑖 where the sum on the right is taken over all 𝑖 such that the 𝑖th 𝑇 component 𝑠𝑖 of 𝑠∈𝑉(𝑛,𝑞)is positive. for 𝑞×𝑞matrices 𝐺1,𝐺2.With𝐺1 =𝐺, 𝐺2 = 𝐺 and thus 𝑇 𝐺1𝐺2 =𝐺𝐺 =𝑞𝐼𝑞, Theorem 7 becomes a special case of this Proof. We note that we have 𝐿 (𝐺𝑇)=𝐿 (𝐺) 𝑠 −1 equation by noting that 𝑠,𝑡 𝑡,𝑠 (see [13,Equation 𝑠 𝑞−1 𝑖 𝑛 𝑞−1 𝑖 { (4.5)]) and 𝐿𝑝,𝑡(𝑞𝐼𝑞)=(𝑞/𝑝!)𝛿𝑝,𝑡 (using the notation of 𝜕 {𝑠 𝑔 (∑𝑔 𝑧 ) , 𝑠 >0, (∑𝑔 𝑧 ) = 𝑖 𝑖𝑘 𝑖𝑗 𝑗 if 𝑖 Definition, 2 the polynomial on the left can only be nonzero 𝑖𝑗 𝑗 { 𝑗=0 𝜕𝑧𝑘 𝑗=0 { if 𝑟𝑖,𝑗 =0whenever 𝑖 =𝑗̸ and 𝑝𝑖 =𝑟𝑖,𝑖 =𝑡𝑖 otherwise). {0, if 𝑠𝑖 =0. (24) 4. Recurrence Relations and Summation of Thus, on the one hand, the derivative of the left-hand side of Certain Polynomials (7)withrespectto𝑧𝑘 is

𝑠 We use Theorem 3 to derive a recurrence relation for 𝑚- 𝑞−1 𝑞−1 𝑠𝑚−1 𝑞−1 𝑞−1 𝑖 polynomials in 𝑠 having parameter 𝑝 where both 𝑠, 𝑝 ∈ [ ] ∑ [𝑠𝑚𝑔𝑚𝑘 (∑𝑔𝑚ℓ𝑧ℓ) ∏(∑𝑔𝑖𝑗 𝑧𝑗) ] . (25) 𝑉(𝑛, 𝑞) by a sum of 𝑚-polynomials where both 𝑠, 𝑝 ∈ 𝑉(𝑛 − 𝑚=0, ℓ=0 𝑖=0, 𝑗=0 𝑠 >0 [ 𝑖 =𝑚̸ ] 1, 𝑞). 𝑚 Using (7), this is equal to Theorem 12 (recurrence relations I). Let 𝑝, 𝑠 ∈ 𝑉(𝑛, 𝑞) with 𝑠𝑖 >0for one 𝑖∈{0,...,𝑞−1}.Then, 𝑞−1 ∑ [𝑠 𝑔 ∑ 𝑀𝐺 (𝑝;̃ 𝑠 −𝑒 )𝑧𝑝̃] . 𝑚 𝑚𝑘 𝑚 (26) 𝑚=0, ̃ 𝑞−1 [ 𝑝∈𝑉(𝑛−1,𝑞) ] 𝑠𝑚>0 𝑀𝐺 (𝑝; 𝑠)= ∑ 𝑔 𝑀𝐺 (𝑝 −𝑒 ;𝑠−𝑒), 𝑖𝑘 𝑘 𝑖 (21) 𝑘=0, On the other hand, the derivative of the right-hand side of (7) 𝑝 >0 𝑘 with respect to 𝑧𝑘 is 𝑝−𝑒 ∑ 𝑝 𝑀𝐺 (𝑝; 𝑠)𝑧 𝑘 . 𝑘 𝑘 𝑘 where the sum on the right is taken over all such that the th 𝑝∈𝑉(𝑛,𝑞), (27) component 𝑝𝑘 of 𝑝∈𝑉(𝑛,𝑞)is positive. 𝑝𝑘>0 ISRN Applied Mathematics 5

Equating the last two expressions and interchanging the order Corollary 15. Let 𝑝, 𝑠 ∈ 𝑉(𝑛, 𝑞) with 𝑠𝑖 ≥𝑚𝑖 ≥0for all 𝑖∈ of summation yield {0,...,𝑞−1}.Then,

𝑞−1 𝑚 𝑀𝐺 (𝑝; 𝑠)= ∑ [∏ ( 𝑖 )] 𝑞−1 𝑟 [ 𝑝̃] (𝑖) 𝑟(𝑖)∈𝑉(𝑚𝑖,𝑞),s.t. 𝑖=0 ∑ [ ∑ 𝑠𝑚𝑔𝑚𝑘𝑀𝐺 (𝑝;̃ 𝑠𝑚 −𝑒 )𝑧 ] 𝑝−∑𝑞−1 𝑟∈𝑉(𝑛−|𝑚|,𝑞) 𝑝∈𝑉(𝑛−1,𝑞)̃ 𝑚=0, 𝑖=0 [𝑠𝑚>0 ] (28) 𝑞−1 𝑞−1 𝑞−1 𝑝−𝑒 𝑟 = ∑ 𝑝 𝑀𝐺 (𝑝; 𝑠)𝑧 𝑘 ; [ 𝑖𝑗 ] 𝑘 × ∏∏𝑔𝑖𝑗 𝑀𝐺 (𝑝 − ∑𝑟(𝑖);𝑠−𝑚), 𝑝∈𝑉(𝑛,𝑞), 𝑖=0 𝑗=0 𝑖=0 𝑝 >0 [ ] 𝑘 (32) 𝑞−1 by comparing coefficients, we must have 𝑝=𝑝−𝑒̃ 𝑘,and where the sum on the right is taken over all the vectors 𝑟 ∈𝑉(𝑚,𝑞) 𝑝−∑𝑞−1 𝑟 ∈ 𝑉(𝑛 − |𝑚|, 𝑞) therefore (noting that 𝑝𝑘 >0) (𝑖) 𝑖 such that 𝑖=0 .

Proof. This follows from repeatedly applying Proposition 14, 𝑞−1 𝑠 𝑚 𝑠 𝑚 𝑠 𝑀𝐺 (𝑝; 𝑠)= ∑ 𝑚 𝑔 𝑀𝐺 (𝑝 −𝑒 ;𝑠−𝑒 ). by subtracting 0 from 0, 1 from 1,andsoon. 𝑝 𝑚𝑘 𝑘 𝑚 (29) 𝑚=0, 𝑘 We note that taking 𝑚=𝑠in the previous corollary yields 𝑠𝑚>0 the statement of Definition 2 (since, by definition, 𝑀𝐺(0; 0) = 1). Combining the recurrence relation in Theorem 12 with Theorem 7 yields the following summation formula for 𝑚- We note that (21)and(23)canberewrittenas polynomials.

𝑟 𝑟 𝑟 Proposition 16 (summation formula). Let 𝐺 be a generalized 𝑀𝐺 (𝑝; 𝑠)= ∑ 𝑔 0 𝑔 1 ⋅⋅⋅𝑔 𝑞−1 𝑖0 𝑖1 𝑖,𝑞−1 Hadamard matrix and 𝑝, 𝑠 ∈ 𝑉(𝑛,. 𝑞) Then, for any pair of 𝑟∈𝑉(1,𝑞),s.t. 𝑝−𝑟∈𝑉(𝑛−1,𝑞) fixed, arbitrary elements 𝑖,𝑗∈{0,...,𝑞−1},onehas 1 𝑞𝑛 ×𝑀𝐺(𝑝−𝑟;𝑠−𝑒𝑖), ∑ 𝑀𝐺 (𝑝 +𝑒 ;𝑠+𝑒) 𝑀𝐺 (𝑝; 𝑠) =𝑔 . 𝑠! 𝑗 𝑖 𝑖𝑗 𝑝! (33) 𝑟 𝑟 𝑟 𝑠∈𝑉(𝑛,𝑞) 𝑠 0 𝑠 1 ⋅⋅⋅𝑠 𝑞−1 0 1 𝑞−1 𝑟 𝑟 𝑟 𝑀𝐺 (𝑝; 𝑠)= ∑ 𝑔 0 𝑔 1 ⋅⋅⋅𝑔 𝑞−1 𝑝 0𝑘 1𝑘 𝑞−1,𝑘 𝑟∈𝑉(1,𝑞),s.t. 𝑘 5. Expansion of a Polynomial 𝑠−𝑟∈𝑉(𝑛−1,𝑞) The coefficients for the expansion of a polynomial in terms ×𝑀𝐺(𝑝−𝑒 ;𝑠−𝑟), 𝑘 of Krawtchouk polynomials have been obtained (cf. [6]). (30) We obtain a similar result for 𝑚-polynomials defined in terms of a generalized Hadamard matrix, using orthogonality respectively. properties. In particular, Theorem 12 canbeeasilyiteratedtoobtain Theorem 17 (polynomial expansion). Let 𝐺 be a generalized the following statements. Hadamard matrix, and let 𝑥=(𝑥0,𝑥1,...,𝑥𝑞−1) be a variable element of 𝑉(𝑛,. 𝑞) Proposition 14. Let 𝑝, 𝑠 ∈ 𝑉(𝑛, 𝑞) with 𝑠𝑖 ≥𝑚≥0for one 𝑖∈{0,...,𝑞−1}.Then, (i) If 𝐺 is symmetric and the expansion of a polynomial 𝛾(𝑥) in terms of 𝑚-polynomials defined with respect to 𝐺 𝛼 ∈ C 𝑚 𝑟 𝑟 𝑟 is (with coefficients 𝑠 ) 𝑀𝐺 (𝑝; 𝑠)= ∑ ( )𝑔0 𝑔 1 ⋅⋅⋅𝑔 𝑞−1 𝑟 𝑖0 𝑖1 𝑖,𝑞−1 𝑟∈𝑉(𝑚,𝑞),s.t. 𝛾 (𝑥) = ∑ 𝛼𝑠 ⋅𝑀𝐺(𝑥;) 𝑠 , 𝑝−𝑟∈𝑉(𝑛−𝑚,𝑞) (31) 𝑠∈𝑉(𝑛,𝑞) (34)

×𝑀𝐺(𝑝−𝑟;𝑠−𝑚𝑒𝑖), then, for all ℓ∈𝑉(𝑛,𝑞), 𝛼 =𝑞−𝑛 ∑ 𝛾 (𝑖) ⋅ 𝑀𝐺 (ℓ, 𝑖). 𝑟∈𝑉(𝑚,𝑞) ℓ (35) where the sum on the right is taken over all such 𝑖∈𝑉(𝑛,𝑞) that 𝑝−𝑟is in 𝑉(𝑛 − .𝑚, 𝑞) Similarly, if the expansion of a polynomial 𝛾(𝑥) in terms Proof. This follows by repeatedly applying (21)inTheorem 12. 𝑚 𝛽𝑠 ∈ C 𝑚 of -polynomials is (with coefficients ) The multinomial coefficient ( 𝑟 ) arisesasthenumberofways 𝑝 canbechangedto𝑝−𝑟step-by-step. 𝛾 (𝑥) = ∑ 𝛽𝑠 ⋅𝑀𝐺(𝑠; 𝑥) , 𝑠∈𝑉(𝑛,𝑞) (36) 6 ISRN Applied Mathematics

then, for all ℓ∈𝑉(𝑛,𝑞), {j, 1, q}])̂(s[[i]]), {i, 1, q}]] 𝛽 =𝑞−𝑛 ∑ 𝛾 (𝑖) ⋅ 𝑀𝐺 (𝑖, ℓ). > mg[p ,s]:= Coefficient[generator[s], ℓ (37) 𝑖∈𝑉(𝑛,𝑞) Inner[Power, z, p, Times]]

(ii) If 𝐺 has a symmetric core, 𝑔0,𝑗 =1for 𝑗 = 0,...,𝑞−1 𝑚 and 𝑔𝑖,0 =−1for 𝑖=1,...,𝑞−1, then the corresponding The values of the -polynomial can be shown using the coefficients can be calculated for all ℓ∈𝑉(𝑛,𝑞)by following command:

ℓ −𝑛 𝑖 𝛼 = (−1) 0 𝑞 ∑ (−1) 0 𝛾 (𝑖) ⋅ 𝑀𝐺 (ℓ, 𝑖), > TableForm[Table[mg[pall[[i]], ℓ (38) 𝑖∈𝑉(𝑛,𝑞) pall[[j]]], {i, Length[pall]}, respectively, {j, Length[pall]}], TableHeadings -> ℓ −𝑛 𝑖 𝛽 = (−1) 0 𝑞 ∑ (−1) 0 𝛾 (𝑖) ⋅ 𝑀𝐺 (𝑖, ℓ). ℓ (39) 𝑖∈𝑉(𝑛,𝑞) {pall, pall}, TableDepth -> 2]

𝑥=𝑖 𝑀𝐺(ℓ, 𝑖) For the above chosen matrix 𝐺 and length 𝑛,theoutput Proof. Let . Multiplying both sides of (34)by 𝑀𝐺(𝑝; 𝑠) 𝑝 𝑠 and summing each side over all elements of 𝑉(𝑛, 𝑞) yield for looks as follows ( denotes the row and the column; e.g., 𝑀𝐺((2, 4); (3, 3))): =3 ∑ 𝛾 (𝑖) 𝑀𝐺 (ℓ, 𝑖) = ∑ ∑ 𝛼𝑠 𝑀𝐺 (ℓ, 𝑖) 𝑀𝐺 (𝑖, 𝑠) 𝑖∈𝑉(𝑛,𝑞) 𝑖∈𝑉(𝑛,𝑞) 𝑠∈𝑉(𝑛,𝑞) {0, 6}{1, 5}{2, 4}{3, 3}{4, 2}{5, 1}{6, 0} {0, 6} 1−11−11−11 = ∑ 𝛼𝑠 ∑ 𝑀𝐺 (𝑙,) 𝑖 𝑀𝐺 (𝑖, 𝑠) {1, 5} −64−202−46 𝑠∈𝑉(𝑛,𝑞) 𝑖∈𝑉(𝑛,𝑞) {2, 4} 15 −5 −1 3 −1 −5 15 {3, 3} −20 0 4 0 −4 0 20 (41) = ∑ 𝛼𝑠 ∑ 𝑀𝐺 (𝑙,) 𝑖 𝑀𝐺 (𝑖, 𝑠) {4, 2} 15 5 −1 −3 −1 5 15 𝑠∈𝑉(𝑛,𝑞) 𝑖∈𝑉(𝑛,𝑞) {5, 1} −6 −4 −2 0 2 4 6 {6, 0} 1111111 (⋆) 𝑛 = ∑ 𝛼𝑠 ⋅𝑞 ⋅𝛿ℓ,𝑠 𝑠∈𝑉(𝑛,𝑞) Thus,weareabletoevaluatethe𝑚-polynomial 𝑀𝐺(𝑝; 𝑠) 𝑛 for any 𝑝, 𝑠 ∈ 𝑉(𝑛, 𝑞) (and thus check orthogonality and =𝛼ℓ 𝑞 , recurrence relations). With the help of the Mathematica (40) function Fit, we can express the m-polynomials as univariant (𝑠 −𝑠) where step (⋆) uses Theorem 10; the first result in (i) follows. polynomials in 0 1 as follows (we also recall that |(𝑠 ,𝑠 )| = 𝑠 +𝑠 =6 Theotherresultsareprovedsimilarly. 0 1 0 1 here): 1 5 𝑀𝐺 ( 0, 6) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 )6 − (𝑠 −𝑠 )4 6. Mathematica Code 0 1 720 0 1 72 0 1 𝑚 Here,wepresentMathematicacodetoobtain -polynomials. 34 2 𝐺 𝑛 + (𝑠 −𝑠 ) −1, First, we specify the matrix and the length : 45 0 1 > g={{1, 1}, {1, −1}} 1 𝑀𝐺 ( 1, 5) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 )5 > n=6 0 1 120 0 1 From this we can calculate 𝑞,theset𝑉(𝑛, 𝑞) and initialize the 1 3 11 − (𝑠0 −𝑠1) + (𝑠0 −𝑠1), variable 𝑧: 3 5 1 7 > q = Union[Dimensions[g]][[1]] 𝑀𝐺 ( 2, 4) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 )4 − (𝑠 −𝑠 )2 +3, 0 1 24 0 1 6 0 1 > pall = Sort[Flatten[Map[Permutations, 1 8 IntegerPartitions[n, {q}, 𝑀𝐺 ( 3, 3) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 )3 − (𝑠 −𝑠 ), 0 1 6 0 1 3 0 1 Range[0, n]]], 1]] 1 > z = Table[f[i], {i, q}] 𝑀𝐺 ( 4, 2) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 )2 −3, 0 1 2 0 1 Using Theorem 3,weobtainthe𝑚-polynomial via its gener- 𝑀𝐺 ( 5, 1) ;(𝑠 ,𝑠 )) = (𝑠 −𝑠 ), ator: 0 1 0 1

> generator[s ]:= Expand[Product[ 𝑀𝐺 ( 6, 0) ;(𝑠0,𝑠1)) = 1. (Sum[g[[i, j]] z[[j]], (42) ISRN Applied Mathematics 7

Similarly, we may express the 𝑚-polynomials as univariant giving the list of coefficients 𝛼𝑠,respectively,𝛽𝑠 for 𝑠∈𝑉(𝑛,𝑞). polynomials in (𝑝0 −𝑝1) as follows: That is, the expansion in terms of 𝑚-polynomials here reads as follows: 77 6 203 4 𝑀𝐺 ((𝑝0,𝑝1);(0, 6))= (𝑝0 −𝑝1) − (𝑝0 −𝑝1) 3 3840 192 𝑥 𝑥 =− 𝑀𝐺 ((𝑥 ,𝑥 );(0, 6)) 0 1 64 0 1 1519 2 + (𝑝0 −𝑝1) − 20, 39 120 − 𝑀𝐺 ((𝑥 ,𝑥 );(2, 4)) 64 0 1 7 5 49 3 𝑀𝐺 ((𝑝0,𝑝1);(1, 5))= (𝑝0 −𝑝1) − (𝑝0 −𝑝1) 7 640 96 + 𝑀𝐺 ((𝑥 ,𝑥 );(4, 2)) 64 0 1 131 (45) + (𝑝0 −𝑝1), 35 30 + 𝑀𝐺 ((𝑥 ,𝑥 );(6, 0)), 64 0 1 7 6 7 4 𝑀𝐺 ((𝑝0,𝑝1);(2, 4))=− (𝑝0 −𝑝1) + (𝑝0 −𝑝1) 1 3840 64 𝑥 𝑥 =− 𝑀𝐺 ( 4, 2) ;(𝑥 ,𝑥 )) 0 1 2 0 1 199 2 − (𝑝0 −𝑝1) +4, 15 120 + 𝑀𝐺 ( 6, 0) ;(𝑥 ,𝑥 )) . 2 0 1 7 5 𝑀𝐺 ((𝑝0,𝑝1);(3, 3))=− (𝑝0 −𝑝1) 1920 As a check, we have (recall that 𝑥0 +𝑥1 =6) 19 3 67 1 15 + (𝑝0 −𝑝1) − (𝑝0 −𝑝1), − 𝑀𝐺 ( 4, 2) ;(𝑥 ,𝑥 )) + 𝑀𝐺 ( 6, 0) ;(𝑥 ,𝑥 )) 96 30 2 0 1 2 0 1

7 6 25 4 1 1 2 15 𝑀𝐺 ((𝑝0,𝑝1);(4, 2))= (𝑝0 −𝑝1) − (𝑝0 −𝑝1) =− ⋅( (𝑥 −𝑥 ) −3)+ ⋅1 (46) 11520 576 2 2 0 1 2

329 2 + (𝑝 −𝑝 ) −4, =𝑥0 (6 − 𝑥0)= 𝑥0𝑥1. 360 0 1 1 For a generalized Hadamard matrix with symmetric core, 𝑀𝐺 ((𝑝 ,𝑝 );(5, 1))= (𝑝 −𝑝 )5 𝑔 =1 𝑗 = 0,...,𝑞−1 𝑔 =−1 𝑖 = 1,...,𝑞−1 0 1 384 0 1 0,𝑗 for and 𝑖,0 for ; for example, 17 19 − (𝑝 −𝑝 )3 + (𝑝 −𝑝 ), 96 0 1 6 0 1 11 1 1−√3 1+√3 1 6 23 4 𝐺=(−1 ). 𝑀𝐺 ((𝑝0,𝑝1);(6, 0))=− (𝑝0 −𝑝1) + (𝑝0 −𝑝1) 2 2 (47) 2304 576 1+√3 1−√3 −1 101 2 2 − (𝑝 −𝑝 )2 + 20. 72 0 1 (43) The above mathematica code using Theorem 17 has to be modified as follows: By Theorem 17, we can express any polynomial 𝛾(𝑥) = > Table[1/q̂n Total [Table 𝛾(𝑥0,...,𝑥𝑞−1) in terms of 𝑚-polynomials. Using Mathemat- ̂ ica to find the corresponding coefficients 𝛼𝑠 and 𝛽𝑠 in the [(-1) (pall[[i, 1]] + pall[[j, 1]]) expansion in terms of 𝑚-polynomials, for example, for the p[pall[[i]]] Conjugate[mg[pall[[j]], polynomial 𝑝(𝑥0,𝑥1)=𝑥0𝑥1, is achieved as follows: pall[[i]]]], > { } p[ x0 ,x1 ]:= x0 x1 {i, Length[pall]}]], {j, Length[pall]}] > ̂ Table[1/q n Total[Table[p[pall[[i]]] > Table[1/q̂n Total[ Conjugate[mg[pall[[j]], pall[[i]]]], Table[(-1)̂(pall[[i, 1]] + pall[[j, 1]]) { } { } i, Length[pall] ]], j, Length[pall] ] p[pall[[i]]] Conjugate[mg[pall[[i]], > Table[1/q̂n Total[Table[p[pall[[i]]] pall[[j]]]], Conjugate[mg[pall[[i]], pall[[j]]]], {i, Length[pall]}]], {j, Length[pall]}] {i, Length[pall]}]], {j, Length[pall]}] The output of the later two lines is 7. Outlook 3 39 7 35 1 15 {− ,0,− ,0, ,0, }, {0,0,0,0,− ,0, }, Krawtchouk polynomials and their generalisation appear 64 64 64 64 2 2 in many areas of Mathematics, see [5]: harmonic analysis (44) [1, 2, 4], statistics [3], combinatorics and coding theory 8 ISRN Applied Mathematics

[6, 7, 9, 11, 14, 15], probability theory [5], representation [14] V. I. Levenshtein, “Krawtchouk polynomials and universal theory (e.g., of quantum groups) [10, 12, 13], difference bounds for codes and designs in Hamming spaces,” IEEE equations [16], and pattern recognition [17](awebsitecalled Transactions on Information Theory,vol.41,no.5,pp.1303–1321, the“KrawtchoukPolynomialsHomePage”byV.Zelenkov 1995. at http://orthpol.narod.ru/eng/index.html collecting material [15] F.J.MacWilliamsandN.J.A.Sloane,Theory of Error-Correcting about M. Krawtchouk and the polynomials that bear his Codes,North-Holland,NewYork,NY,USA,1978. name has, unfortunately, not been updated in a while). How- [16] L. Boelen, G. Filipuk, C. Smet, W. Van Assche, and L. ever, our motivation in this paper was primarily driven by Zhang, “The generalized Krawtchouk polynomials and the generalising the results obtained in [8]—and thus shedding fifth Painleve equation,” Journal of Difference Equations and more light on the qualitative structure of (generalisations Applications,vol.19,pp.1437–1451,2013. of) Krawtchouk polynomials—and not yet with a specific [17] J. Sheeba Rani and D. Devaraj, “Face recognition using application in mind. By providing the Mathematica code to Krawtchouk moment,” Sadhan¯ a¯,vol.37,no.4,pp.441–460, obtain 𝑚-polynomials, we also hope that other researchers 2012. areencouragedtoexplorethemandseeiftheycanbeusedin their research.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] M. Krawtchouk, “Sur une gen´ eralization´ des polynomes d’Hermite,” ComptesRendusdel’Academie´ des Sciences,vol.189, pp. 620–622, 1929. [2] G. Szego, Orthogonal Polynomials, vol. 23, Colloquium, New York, NY, USA, 1959. [3] G. K. Eagleson, “Acharacterization theorem for positive definite sequences on the Krawtchouk polynomials,” The Australian Journal of Statistics,vol.11,pp.29–38,1969. [4] C. F. Dunkl, “A Krawtchouk polynomial addition theorem andwreathproductsofsymmetricgroups,”Indiana University Mathematics Journal,vol.25,no.4,pp.335–358,1976. [5] P. Feinsilver and J. Kocik, “Krawtchouk polynomials and Krawtchouk matrices,” in Recent Advances in Applied Probabil- ity, R. Baeza-Yates, J. Glaz, H. Gzyl, J. Husler,¨ and J. L. Palacios, Eds., pp. 115–141, Springer, New York, NY, USA, 2005. [6]F.J.MacWilliams,N.J.A.Sloane,andJ.-M.Goethals,“The MacWilliams identities for nonlinear codes,” The Bell System Technical Journal,vol.51,pp.803–819,1972. [7] I. Krasikov and S. Litsyn, “On integral zeros of Krawtchouk polynomials,” JournalofCombinatorialTheoryA,vol.74,no.1, pp.71–99,1996. [8] N. Sookoo, “Generalized Krawtchouk polynomials: new prop- erties,” Archivum Mathematicum,vol.36,no.1,pp.9–16,2000. [9] L. Habsieger, “Integer zeros of q-Krawtchouk polynomials in classical combinatorics,” Advances in Applied Mathematics,vol. 27, no. 2-3, pp. 427–437, 2001. [10] M. N. Atakishiyev and V. A. Groza, “The quantum algebra 𝑈𝑞(𝑠𝑢2) and 𝑞-Krawtchouk families of polynomials,” Journal of Physics A,vol.37,no.7,pp.2625–2635,2004. [11] P.Delsarte, “Analgebraic approach to the association schemes of coding theory,” Philips Research Reports,no.10,97pages,1973. [12] J. D. Louck, “Group theory of harmonic oscillators in 𝑛- dimensional space,” Journal of Mathematical Physics,vol.6,pp. 1786–1804, 1965. [13]W.Y.C.ChenandJ.D.Louck,“Thecombinatoricsofaclass of representation functions,” Advances in Mathematics,vol.140, no. 2, pp. 207–236, 1998. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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