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 q-Krawtchouk polynomials in [9]. Orthogonality relations for quantum and q-Krawtchouk polynomials have been derived in [10], and it has been shown that affine q-Krawtchouk polynomials are dual to quantum q-Krawtchouk polynomials. In this paper, we define and study generalizations of Krawtchouk polynomials, namely, m-polynomials.
The Krawtchouk polynomial Kk(x) is given by
(1)Kk(x)=∑j=0k(-1)j(xj)(n-xk-j),
where n is a natural number and x∈{0,1,…,n}. The generator polynomial is
(2)(1+z)n-x(1-z)x=∑k=0nKk(x)zk.
The generalized Krawtchouk polynomial Kp(s) is obtained by generalizing the above generator polynomial as follows:
(3)∏l=0q-1(∑i=0q-1χ(ω¯iω¯l)zi)si=∑p∈V(n,q)Kp(s)zp,
where ∑si=∑pi=n, q is a prime power, the zi are indeterminate, the field GF(q) with q elements is {0,ω¯1,ω¯2,…,ω¯q-1}, and χ is a character.
The above information about Krawtchouk polynomials and generalized Krawtchouk polynomials was taken from [6].
If we replace the χ(ω¯iω¯l) by arbitrary scalars in the last equation, we obtain the generator polynomial of m-polynomials MG(p,s); see Definition 2 below. These m-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 G can be any square matrix, and so the question that immediately arises is how the properties of the m-polynomials are related to the properties of G. We will establish that, if G is a generalized Hadamard matrix, then the associated m-polynomials satisfy orthogonality conditions. In Section 4, we establish recurrence relations for m-polynomials. Afterwards, we obtain coefficients for the expansion of a polynomial in terms of m-polynomials in Section 5. Finally, in Section 6, we implement the results obtained here in Mathematica, so the reader may easily derive and explore m-polynomials for any matrix G.
2. Definitions and Notations
In this paper, ℕ0 denotes {0,1,2,3,…}. We use the convention that if p∈ℕ0q, then p has components (p0,p1,…,pq-1). So, if r(i)∈ℕ0q then r(i) denotes (ri,0,ri,1,…,ri,q-1). We will also use the qelementary unit vectorse0=(1,0,0,…,0), e1=(0,1,0,…,0),…,eq-1=(0,0,…,0,1).
We use the ℓ1-norm (the “taxicab-metric”) to measure the length|p| of p∈ℕ0q, that is, |p|=∑i=0q-1pi. We define the set of weak compositions of n into q numbers by V(n,q)={p∈ℕ0q:|p|=n}; in other words, V(n,q) is the subset of q-dimensional nonnegative vectors of length n. We note that the set V(n,q) has cardinality (n+q-1q-1). For p∈ℕ0q, we use the multi-index notation p!=∏i=0q-1pi!. Similarly, for a variable z=(z0,z1,…,zq-1)∈ℂq and p∈ℕ0q, we write zp=∏j=0q-1zjpi (where the convention 00=1 is used). We note that the multinomial theorem reads
(4)(∑i=0q-1zi)n=∑p∈V(n,q)(np)·zp=∑p∈V(n,q)n!p0!⋯pq-1!·z0p0⋯zq-1pq-1.
In the following, G denotes an arbitrary q×q matrix. We use the following convention to refer to the entries of a q×q matrix [11]. The entry in the ith row and jth column is called the (i-1,j-1)entry, where i,j=1,2,…,q. Thus, the (i,j) entry of G is denoted by gij, where i,j=0,1,…,q-1. Given a matrix G, the matrix that remains when the first row and the first column of G are removed is called the core of G.
The next definition is well known.
Definition 1.
For an integer q greater than one, a q×q matrix H is called a generalized Hadamard matrix if HH¯T=qIq, where H¯T is the complex conjugate transpose of H and Iq is the q×q identity matrix.
We now define the m-polynomial with respect to a matrix G.
Definition 2.
Let s=(s0,s1,,…,sq-1) and p=(p0,p1,…,pq-1) be elements of V(n,q). Them-polynomial in s with respect to G having parameter p is denoted by MG(p;s) or MG(p0,p1,…,pq-1;s0,s1,…,sq-1) and given by
(5)MG(p;s)=∑ri,js!∏i,jri,j!∏a=0q-1∏b=0q-1gabra,b=s!∑ri,j∏a=0q-1∏b=0q-1gabra,bra,b!,
where the summation is taken over all sets of nonnegative integers ri,j (with i,j=0,1,…,q-1) satisfying
(6)∑j=0q-1ri,j=si,i=0,1,…,q-1,∑i=0q-1ri,j=pj,j=0,1,…,q-1.
3. The Generator Polynomial and Orthogonality Relations
The values of the m-polynomial with respect to a matrix G can be derived from a generator polynomial.
Theorem 3 (generator polynomial).
Let s=(s0,s1,…,sq-1)∈V(n,q) and let G=(gij) be a q×q matrix. Then,
(7)∏i=1q-1(∑j=0q-1gijzj)si=∑p∈V(n,q)MG(p;s)zp,
where p=(p0,p1,…,pq-1).
Proof.
This is an application of the multinomial theorem (recall that for r(i)∈ℕ0q, we have r(i)=(ri,0,ri,1,…,ri,q-1)):
(8)∏i=0q-1(∑j=0q-1gijzj)si=∏i=0q-1{∑r(i)∈V(si,q)si!r(i)![∏j=0q-1(gijzj)ri,j]}=∑r(i)∈V(si,q),i=0,1,…,q-1s!∏i,jri,j!(∏a=1q-1∏b=0q-1gabra,b)×∏k=0q-1zkr0,k+r1,k+⋯+rq-1,k=∑p∈V(n,q)MG(p;s)zp,
where pk=r0,k+r1,k+⋯+rq-1,k.
As an immediate consequence, we can recover the entries gij of the matrix G as the m-polynomials of minimum order.
Corollary 4.
We have MG(eℓ;ek)=gkℓ.
Proof.
For s=ek, the left-hand side of (7) becomes
(9)gk,0z0+gk,1z1+⋯+gk,q-1zq-1,
and since p∈V(1,q) and thus runs through e0,…,eq-1 in this case, the rest follows by comparing coefficients of zj.
Remark 5.
Theorem 3 can also be used for summation results of m-polynomials over V(n,q): using z=(1,…,1), we obtain
(10)∑p∈V(n,q)MG(p;s)=∏i=0q-1(∑j=0q-1gij)si,
that is, the product of the sith power of the ith column sum of G.
As an immediate result of this remark, we can establish the following corollary.
Corollary 6.
If ∑j=0q-1gij=0 for one 0≤i≤q-1 (i.e., one of the row-sums of the matrix G is zero), then ∑p∈V(n,q)MG(p;s)=0.
For a generalized Hadamard matrix G, the multinomial theorem yields the following orthogonality relation for the corresponding m-polynomials.
Theorem 7 (orthogonality relation).
If G is a generalized Hadamard matrix, then the m-polynomials MG(p,s), with p,s∈V(n,q), satisfy the orthogonality relations
(11)∑s∈V(n,q)1s!MG(p;s)MG(t;s)¯=qnp!δp,t,
where
(12)s=(s0,s1,…,sq-1),p=(p0,p1,…,pq-1),t=(t0,t1,…,tq-1),δp,t=∏i=0q-1δpi,ti={1,ifpi=ti∀i,0,otherwise,
denotes Kronecker’s delta.
Proof.
Let z=(z0,z1,…,zq-1), y=(y0,y1,…,yq-1)∈ℂq, and define
(13)J=∑i=0q-1(∑j=0q-1gijzj)(∑k=0q-1gik¯yk)=∑i=0q-1∑j=0q-1∑k=0q-1gijgik¯zjyk.
Then, on the one hand, we have
(14)J=q∑j=0q-1zjyj
since G is a generalized Hadamard matrix. Consequently,
(15)Jn=qn{∑s∈V(n,q)n!s!(z0y0)s0(z1y1)s1⋯(zq-1yq-1)sq-1}.
On the other hand, the multinomial theorem yields
(16)Jn=∑s∈V(n,q)n!s!∏i=0q-1(∑j=0q-1gijzj)si(∑k=0q-1gik¯yk)si=∑s∈V(n,q)n!s![∑p∈V(n,q)MG(p;s)zp]×[∑t∈V(n,q)MG(t;s)¯yt].
Equating coefficients of zpyt in the two above expressions for Jn, we obtain the desired result.
If G is a generalized Hadamard matrix and also satisfies certain additional conditions, then it is possible to establish that the corresponding m-polynomials satisfy additional orthogonality conditions. We use the following three results in proving this.
Lemma 8.
If G is symmetric, that is, G=GT, then MG(s;p)=(p!/s!)MG(p;s).
Proof.
By Definition 2, we have (where ∑j=0q-1hi,j=pi and ∑i=0q-1hi,j=sj)
(17)MG(s;p)=p!∑hi,j∏a=0q-1∏b=0q-1gabha,bha,b!=p!s!s!∑hi,j∏a=0q-1∏b=0q-1gbaha,bha,b!=p!s!s!∑ri,j∏b=0q-1∏a=0q-1gbarb,arb,a!=p!s!MG(p;s),
where we define ri,j=hj,i (and thus ∑j=0q-1ri,j=si and ∑i=0q-1ri,j=pj), and since G is symmetric (i.e., gab=gba).
The following results can be obtained in a similar manner to Lemma 8.
Lemma 9.
If G has a symmetric core, g0,j=1 for j=0,…,q-1 and gi,0=-1 for i=1,…,q-1, then MG(s;p)=(-1)s0+p0(p!/s!)MG(p;s).
Such symmetry relations yield additional orthogonality relations.
Theorem 10 (additional orthogonality relation).
Let G be a generalized Hadamard matrix.
If in addition G is symmetric, then (18)∑s∈V(n,q)MG(p;s)MG(s;t)¯=qnδp,t.
If in addition G has a symmetric core, g0,j=1 for j=0,…,q-1 and gi,0=-1 for i=1,…,q-1, then (19)∑s∈V(n,q)(-1)s0MG(p;s)MG(s;t)¯=(-1)t0qnδp,t.
Proof.
This follows immediately from Theorem 7 and Lemmas 8 and 9.
Remark 11.
It turns out that a slight modification of the above m-polynomials has already been considered in [12, 13]. For a matrix G and p,s∈V(n,q), [13, Equation (2.3)] defines polynomials Lp,s(G) by Lp,s(G)=(1/s!)MG(p;s) in our notation. For these polynomials, the following multiplication property is proved (see [13, Theorem 3.2]):
(20)Lp,t(G1G2)=∑s∈V(n,q)s!Lp,s(G1)Ls,t(G2),
for q×q matrices G1, G2. With G1=G, G2=G¯T and thus G1G2=GG¯T=qIq, Theorem 7 becomes a special case of this equation by noting that Ls,t(GT)=Lt,s(G) (see [13, Equation (4.5)]) and Lp,t(qIq)=(qn/p!)δp,t (using the notation of Definition 2, the polynomial on the left can only be nonzero if ri,j=0 whenever i≠j and pi=ri,i=ti otherwise).
4. Recurrence Relations and Summation of Certain Polynomials
We use Theorem 3 to derive a recurrence relation for m-polynomials in s having parameter p where both s,p∈V(n,q) by a sum of m-polynomials where both s,p∈V(n-1,q).
Theorem 12 (recurrence relations I).
Let p,s∈V(n,q) with si>0 for one i∈{0,…,q-1}. Then,
(21)MG(p;s)=∑k=0,pk>0q-1gikMG(p-ek;s-ei),
where the sum on the right is taken over all k such that the kth component pk of p∈V(n,q) is positive.
Proof.
Using (7) twice, we obtain (if si>0)
(22)∑p∈V(n,q)MG(p;s)zp=∏i=1q-1(∑j=0q-1gijzj)si=(∑j=0q-1gijzj)(∑j=0q-1g0jzj)s0⋯(∑j=0q-1gi-1,jzj)si-1,(∑j=0q-1gijzj)si-1(∑j=0q-1gi+1,jzj)si+1⋯(∑j=0q-1gq-1,jzj)sq-1=(∑j=0q-1gijzj)∑p~∈V(n-1,q)MG(p~;s-ei)zp~.
Multiplying the right-hand side out and equating coefficients of zp establish the claim.
Theorem 13 (recurrence relations II).
Let p,s∈V(n,q) with pk>0 for one k∈{0,…,q-1}. Then,
(23)MG(p;s)=∑i=0,si>0q-1sipkgikMG(p-ek;s-ei),
where the sum on the right is taken over all i such that the ith component si of s∈V(n,q) is positive.
Proof.
We note that we have
(24)∂∂zk(∑j=0q-1gijzj)si={sigik(∑j=0q-1gijzj)si-1,ifsi>0,0,ifsi=0.
Thus, on the one hand, the derivative of the left-hand side of (7) with respect to zk is
(25)∑m=0,sm>0q-1[smgmk(∑ℓ=0q-1gmℓzℓ)sm-1∏i=0,i≠mq-1(∑j=0q-1gijzj)si].
Using (7), this is equal to
(26)∑m=0,sm>0q-1[smgmk∑p~∈V(n-1,q)MG(p~;s-em)zp~].
On the other hand, the derivative of the right-hand side of (7) with respect to zk is
(27)∑p∈V(n,q),pk>0pkMG(p;s)zp-ek.
Equating the last two expressions and interchanging the order of summation yield
(28)∑p~∈V(n-1,q)[∑m=0,sm>0q-1smgmkMG(p~;s-em)zp~]=∑p∈V(n,q),pk>0pkMG(p;s)zp-ek;
by comparing coefficients, we must have p~=p-ek, and therefore (noting that pk>0)
(29)MG(p;s)=∑m=0,sm>0q-1smpkgmkMG(p-ek;s-em).
We note that (21) and (23) can be rewritten as
(30)MG(p;s)=∑r∈V(1,q),s.t.p-r∈V(n-1,q)gi0r0gi1r1⋯gi,q-1rq-1gggggggggggggggggggggg×MG(p-r;s-ei),MG(p;s)=∑r∈V(1,q),s.t.s-r∈V(n-1,q)s0r0s1r1⋯sq-1rq-1pkg0kr0g1kr1⋯gq-1,krq-1ggggggggggg×MG(p-ek;s-r),
respectively.
In particular, Theorem 12 can be easily iterated to obtain the following statements.
Proposition 14.
Let p,s∈V(n,q) with si≥m≥0 for one i∈{0,…,q-1}. Then,
(31)MG(p;s)=∑r∈V(m,q),s.t.p-r∈V(n-m,q)(mr)gi0r0gi1r1⋯gi,q-1rq-1gggggggggg×MG(p-r;s-mei),
where the sum on the right is taken over all r∈V(m,q) such that p-r is in V(n-m,q).
Proof.
This follows by repeatedly applying (21) in Theorem 12. The multinomial coefficient (mr) arises as the number of ways p can be changed to p-r step-by-step.
Corollary 15.
Let p,s∈V(n,q) with si≥mi≥0 for all i∈{0,…,q-1}. Then,
(32)MG(p;s)=∑r(i)∈V(mi,q),s.t.p-∑i=0q-1r∈V(n-|m|,q)[∏i=0q-1(mir(i))]×[∏i=0q-1∏j=0q-1gijrij]MG(p-∑i=0q-1r(i);s-m),
where the sum on the right is taken over all the q-1 vectors r(i)∈V(mi,q) such that p-∑i=0q-1r∈V(n-|m|,q).
Proof.
This follows from repeatedly applying Proposition 14, by subtracting m0 from s0, m1 from s1, and so on.
We note that taking m=s in the previous corollary yields 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-polynomials.
Proposition 16 (summation formula).
Let G be a generalized Hadamard matrix and p,s∈V(n,q). Then, for any pair of fixed, arbitrary elements i,j∈{0,…,q-1}, one has
(33)∑s∈V(n,q)1s!MG(p+ej;s+ei)MG(p;s)¯=gijqnp!.
5. 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 m-polynomials defined in terms of a generalized Hadamard matrix, using orthogonality properties.
Theorem 17 (polynomial expansion).
Let G be a generalized Hadamard matrix, and let x=(x0,x1,…,xq-1) be a variable element of V(n,q).
If G is symmetric and the expansion of a polynomial γ(x) in terms of m-polynomials defined with respect to G is (with coefficients αs∈ℂ)
(34)γ(x)=∑s∈V(n,q)αs·MG(x;s),
then, for all ℓ∈V(n,q),
(35)αℓ=q-n∑i∈V(n,q)γ(i)·MG(ℓ,i)¯.
Similarly, if the expansion of a polynomial γ(x) in terms of m-polynomials is (with coefficients βs∈ℂ)
(36)γ(x)=∑s∈V(n,q)βs·MG(s;x),
then, for all ℓ∈V(n,q),
(37)βℓ=q-n∑i∈V(n,q)γ(i)·MG(i,ℓ)¯.
If G has a symmetric core, g0,j=1 for j=0,…,q-1 and gi,0=-1 for i=1,…,q-1, then the corresponding coefficients can be calculated for all ℓ∈V(n,q) by
(38)αℓ=(-1)ℓ0q-n∑i∈V(n,q)(-1)i0γ(i)·MG(ℓ,i)¯,
respectively,
(39)βℓ=(-1)ℓ0q-n∑i∈V(n,q)(-1)i0γ(i)·MG(i,ℓ)¯.
Proof.
Let x=i. Multiplying both sides of (34) by MG(ℓ,i)¯ and summing each side over all elements of V(n,q) yield
(40)∑i∈V(n,q)γ(i)MG(ℓ,i)¯=∑i∈V(n,q)∑s∈V(n,q)αsMG(ℓ,i)¯MG(i,s)=∑s∈V(n,q)αs∑i∈V(n,q)MG(l,i)¯MG(i,s)=∑s∈V(n,q)αs∑i∈V(n,q)MG(l,i)MG(i,s)¯¯=(⋆)∑s∈V(n,q)αs·qn·δℓ,s=αℓqn,
where step (⋆) uses Theorem 10; the first result in (i) follows. The other results are proved similarly.
6. Mathematica Code
Here, we present Mathematica code to obtain m-polynomials. First, we specify the matrix G and the length n:
> g = {{1, 1}, {1, −1}}
> n = 6
From this we can calculate q, the set V(n,q) and initialize the variable z:
> q = Union[Dimensions[g]][[1]]
> pall = Sort[Flatten[Map[Permutations,
IntegerPartitions[n, {q},
Range[0, n]]], 1]]
> z = Table[f[i], {i, q}]
Using Theorem 3, we obtain the m-polynomial via its generator:
> generator[s_]:= Expand[Product[
(Sum[g[[i, j]] z[[j]],
{j, 1, q}])^(s[[i]]), {i, 1, q}]]
> mg[p_, s_]:= Coefficient[generator[s],
Inner[Power, z, p, Times]]
The values of the m-polynomial can be shown using the following command:
> TableForm[Table[mg[pall[[i]],
pall[[j]]], {i, Length[pall]},
{j, Length[pall]}], TableHeadings ->
{pall, pall}, TableDepth -> 2]
For the above chosen matrix G and length n, the output for MG(p;s) looks as follows (p denotes the row and s the column; e.g., MG((2,4);(3,3))=3):
(41){0,6}{1,5}{2,4}{3,3}{4,2}{5,1}{6,0}{0,6}1-11-11-11{1,5}-64-202-46{2,4}15-5-13-1-515{3,3}-20040-4020{4,2}155-1-3-1515{5,1}-6-4-20246{6,0}1111111
Thus, we are able to evaluate the m-polynomial MG(p;s) for any p,s∈V(n,q) (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 (s0-s1) as follows (we also recall that |(s0,s1)|=s0+s1=6 here):
(42)MG((0,6);(s0,s1))=1720(s0-s1)6-572(s0-s1)4+3445(s0-s1)2-1,MG((1,5);(s0,s1))=1120(s0-s1)5-13(s0-s1)3+115(s0-s1),MG((2,4);(s0,s1))=124(s0-s1)4-76(s0-s1)2+3,MG((3,3);(s0,s1))=16(s0-s1)3-83(s0-s1),MG((4,2);(s0,s1))=12(s0-s1)2-3,MG((5,1);(s0,s1))=(s0-s1),MG((6,0);(s0,s1))=1.
Similarly, we may express the m-polynomials as univariant polynomials in (p0-p1) as follows:
(43)MG((p0,p1);(0,6))=773840(p0-p1)6-203192(p0-p1)4+1519120(p0-p1)2-20,MG((p0,p1);(1,5))=7640(p0-p1)5-4996(p0-p1)3+13130(p0-p1),MG((p0,p1);(2,4))=-73840(p0-p1)6+764(p0-p1)4-199120(p0-p1)2+4,MG((p0,p1);(3,3))=-71920(p0-p1)5+1996(p0-p1)3-6730(p0-p1),MG((p0,p1);(4,2))=711520(p0-p1)6-25576(p0-p1)4+329360(p0-p1)2-4,MG((p0,p1);(5,1))=1384(p0-p1)5-1796(p0-p1)3+196(p0-p1),MG((p0,p1);(6,0))=-12304(p0-p1)6+23576(p0-p1)4-10172(p0-p1)2+20.
By Theorem 17, we can express any polynomial γ(x)=γ(x0,…,xq-1) in terms of m-polynomials. Using Mathematica to find the corresponding coefficients αs and βs in the expansion in terms of m-polynomials, for example, for the polynomial p(x0,x1)=x0x1, is achieved as follows:
> p[{x0_, x1_}]:= x0 x1
> Table[1/q^n Total[Table[p[pall[[i]]]
Conjugate[mg[pall[[j]], pall[[i]]]],
{i, Length[pall]}]], {j, Length[pall]}]
> Table[1/q^n Total[Table[p[pall[[i]]]
Conjugate[mg[pall[[i]], pall[[j]]]],
{i, Length[pall]}]], {j, Length[pall]}]
The output of the later two lines is
(44){-364,0,-3964,0,764,0,3564},{0,0,0,0,-12,0,152},
giving the list of coefficients αs, respectively, βs for s∈V(n,q). That is, the expansion in terms of m-polynomials here reads as follows:
(45)x0x1=-364MG((x0,x1);(0,6))-3964MG((x0,x1);(2,4))+764MG((x0,x1);(4,2))+3564MG((x0,x1);(6,0)),x0x1=-12MG((4,2);(x0,x1))+152MG((6,0);(x0,x1)).
As a check, we have (recall that x0+x1=6)
(46)-12MG((4,2);(x0,x1))+152MG((6,0);(x0,x1))=-12·(12(x0-x1)2-3)+152·1=x0(6-x0)=x0x1.
For a generalized Hadamard matrix with symmetric core, g0,j=1 for j=0,…,q-1 and gi,0=-1 for i=1,…,q-1; for example,
(47)G=(111-11-321+32-11+321-32).
The above mathematica code using Theorem 17 has to be modified as follows:
> Table[1/q^n Total [Table
[(-1)^(pall[[i, 1]] + pall[[j, 1]])
p[pall[[i]]] Conjugate[mg[pall[[j]],
pall[[i]]]],
{i, Length[pall]}]], {j, Length[pall]}]
> Table[1/q^n Total[
Table[(-1)^(pall[[i, 1]] + pall[[j, 1]])
p[pall[[i]]] Conjugate[mg[pall[[i]],
pall[[j]]]],
{i, Length[pall]}]], {j, Length[pall]}]
7. 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 [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 m-polynomials, we also hope that other researchers are encouraged to explore them and see if they can be used in their research.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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