A conceptual unification is given of these two group theoretic interpretations of krawtchouk polynomials. The proof uses several recurrences and other properties satisfied by the krawtchouk polynomials 17. Hecke algebra, qkrawtchouk polynomial, zonal spherical function. In section 1, we find a model for hahn polynomials and prove the seven arrows of fig. Integer zeros of qkrawtchouk polynomials in classical. Survey of binary krawtchouk polynomials request pdf. An algebraic interpretation of the multivariate q krawtchouk. A problem in potential theory and zero asymptotics of krawtchouk polynomials p. Three addition theorems for some qkrawtchouk polynomials. Attention is drawn to a phenomenon ofpseudostabilityin connection with the threeterm recurrence relation for discrete orthogonal polynomials. On characterizing integral zeros of krawtchouk polynomials by.
On characterizing integral zeros of krawtchouk polynomials. Writing x, y for x0,x1, we have in degree n for the kth. Since the 41 weighted krawtchouk polynomials are discrete, there is no nu42 merical approximation in deriving the transform coef. The method for all these cases is the same and we observe that the method also works for polynomials outside the class of hypergeometric polynomials, such as bernoulli and euler. Reproducing kernel orthogonal polynomials on the multinomial. I have not gone through and given citations or references for all of the results given here, but the presentation relies heavily on two sources, van. A problem in potential theory and zero asymptotics of. Krawtchouk matrices have as entries values of the krawtchouk polynomials for nonnegative integer arguments. Pdf on a family of 2variable orthogonal krawtchouk. The associated polynomials are related to affine matrix groups. Universal bounds for the cardinality of codes in the hamming space fsub rsup n with a given minimum distance d andor dual distance d are stated.
We begin with krawtchouk polynomials for general parameter p and derive an identity for partial sums of squares along a column. In recent years, the study of the asymptotic property for krawtchouk polynomials appears continuously in some important mathematics publications. Krawtchouks polynomial for hypergeometric distribution. In the multivariate krawtchouk polynomials, there is not a. But avoid asking for help, clarification, or responding to other answers. Integral zeroes of krawtchouk polynomials by ahmadm. A partially ordered set and gkrawtchouk polynomials. A detailed study of integral roots of binary krawtchouk polynomials was undertaken in 9, 16. In this case, the series is called a hypergeometric polynomial. Uvarov, special functions of mathematical physics, birkhauser 1988 translated from russian.
This method is used to form a measure algebra whose characters are krawtchouk polynomials. My aim in writing this article is to fill in this gap. Generalizing krawtchouk polynomials using hadamard matrices. Biorthogonal codes with spherically restricted positions. On integral zeros of krawtchouk polynomials semantic scholar. Although the basic properties of these polynomials are to some extent known, there is, to my knowledge, no detailed development available. Using techniques of dunkl to decompose the irreducible representations with respect to a maximal parabolic subgroup, we derive three addition theorems. Koornwinder, krawtchouk polynomials, a unification of two different group theoretic interpretations siam j. Krawtchouk and multivariate krawtchouk polynomial hypergroups. Research article generalizing krawtchouk polynomials using.
Krawtchouk orthogonal polynomials the krawtchouk orthogonal polynomials knx. Dragnev1 department of mathematics, indianapurdue university, fort wayne, indiana 46805, u. In this context, tichy and the author recently showed that a binary diophantine. Kravchuk polynomials or krawtchouk polynomials also written using several other transliterations of the ukrainian name. Krawtchouk polynomials and their generalisation appear in many areas of mathematics, see. In this paper, we stress this viewpoint by giving a new, uniform proof that the qkrawtchouk polynomials are the eigenvalues of these association schemes. Recently, there is a considerable interest in the asymptotics of krawtchouk polynomials, when the degree n grows to infinity.
A uniform asymptotic expansion for krawtchouk polynomials. The necessary and sufficient conditions for attainability of the bounds are found. The integral zeros for two families of q krawtchouk polynomials are classi. The dual multivariate krawtchouk polynomials, which also have a polynomial structure, are seen to occur.
In mathematics, the dual qkrawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic askey scheme. Pdf krawtchouk polynomials, the lie algebra sl 2, and. The nonsymmetric krawtchouk polynomials have received by the editors june 3, 1996 and, in revised form, november 1, 1998. Chami, 1 berndsing, 1 andnorrissookoo 2 department of computer science, mathematics and physics, faculty of science and technology, e university of the west indies, cave hill, st. The multivariate krawtchouk polynomials are extensions of this construction in higher dimensions. Reconstruction problems for graphs, krawtchouk polynomials. Is the recurrence relation for orthogonal polynomials always. Sums of squares of krawtchouk polynomials, catalan numbers. For instance, in 12, sharapudinov has obtained the asymptotic formula. For additional properties of these polynomials, see the references cited in 4, p. Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory.
Thanks for contributing an answer to mathematics stack exchange. A selfcontained proof of optimality of these bounds in the framework of the linear programming method is given. Reconstruction problems for graphs, krawtchouk polynomials and diophantine equations thomas stoll june 11, 2008 abstract we give an overview about some reconstruction problems in graph theory, which are intimately related to integer roots of krawtchouk polynomials. When q 2, these polynomials are called binary krawtchouk polynomials. Krawtchouk polynomials and universal bounds for codes and. We would like to mention several questions which appear to be out of the scope of the paper but very much similar to its. In applications the most important case is the binary krawtchouk polynomials corresponding to q 1 2. Dunkls interpretation of krawtchouk polynomials as spherical functions on wreath products of symmetric groups is generalized to the case of intertwining functions. Let wbe a compact group ofautomorphisms on the compact p,hypergroup h. A short proof of a generating function for jacobi polynomials, proc. Krawtchouk polynomials and the symmetrization of hypergroups. On zeros of discrete orthogonal polynomials sciencedirect.
The multivariate krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of n multinomial trials. Research article generalizing krawtchouk polynomials using hadamard matrices peters. It is our experience, and the experience of many others, that the basic three term recurrence relation for orthogonal polynomials is generally an excellent means. Thus the krawtchouk polynomials will be considered to be the special case. The paper begins with expressing the 1dimensional krawtchouk polynomials as symmetric functions of bernoulli trials. Krawtchouk polynomials encyclopedia of mathematics. Journal of statistical planning and inference 154, 3953. Citeseerx document details isaac councill, lee giles, pradeep teregowda. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. The columns of krawtchouk matrices may be considered generalized binomial coe. Departmentofmathematics bruneluniversity uk october,2012 1. Some qkrawtchouk polynomials on chevalley groups, amer. Ieee transactions on signal processing 1 fractional. The krawtchouk and multivariate krawtchouk polynomials are symmetric functions of orthogonal functions sets on each of the trials.
Retrieved from the university of minnesota digital. They are symmetric with respect to n 2, so x k n 2. The algebraic interpretation is used to derive the main properties of the polynomials. Krawtchouk polynomials, a unification of two different. N vn k nj,n the polynomials knj,n are orthogonal with respect to the binomial distribution with parameters n,p. Next, 1 we construct a set of orthonormal eigenvectors of matrix k 2 corresponding to the two eigenvalues. Theset obis compactin the pointwise topologyfromh,andhence in theweaktopologyas asubset ofch. Krawtchouk polynomials and krawtchouk matrices arxiv. For general properties of roots of krawtchouk polynomials see 26, 40. Kravchuk polynomials article about kravchuk polynomials by. Multivariate krawtchouk polynomials and a spectral theorem.
We give a hypergeometric proof involving a family of 2variable krawtchouk polynomials that were obtained earlier by hoare and rahman sigma 4 2008, 089, 18 pages as a limit of the 9. The result follows from theorem 1 and the binomial theorem. Pdf krawtchouk polynomials and krawtchouk matrices. The integral zeros for two families of qkrawtchouk polynomials are classi. Multivariate krawtchouk polynomials and composition birth and. Krawtchouk polynomials appear originally as orthogonal polynomials for the. We show how they arise as condensed sylvesterhadamard matrices via a binary shuffling function. Hecke algebra, q krawtchouk polynomial, zonal spherical function. The zeros of generalized krawtchouk polynomials are studied. Ieee proof liu et al fractional krawtchouk transform with an application to image watermarking 3 theorem 1 speci. Krawtchouk polynomials appear in a variety of contexts.
The askey scheme for hypergeometric orthogonal polynomials. A simple model for meixnerpollaczek, krawtchouk and meixner polynomials is introduced in section 2 and four more limit formulas fig. Orthogonality relations for multivariate krawtchouk polynomials. A generating function for the symmetric functions on the right of.
The multivariate quantum qkrawtchouk polynomials are shown to arise as matrix elements of qrotations acting on the state vectors of many qoscillators. This hypergroup property leads to a characterization of reversible markov chains with multinomial stationary distributions and multivariate krawtchouk eigenfunctions. Theq krawtchouk polynomials are the spherical functions for three different chevalley groups over a finite field. Krawtchouk polynomials, catalan numbers, super catalan numbers, boolean lattice, algebras, zeons. Saff2 department of mathematics, institute for constructive mathematics, university of south florida, tampa, florida 33620, u. Is the recurrence relation for orthogonal polynomials. The approach of 21 on composition markov chains ties in with the multivariate polynomials as well, cf. Hahn polynomials and multivariate krawtchouk polynomials 16,17,20,35. The krawtchouk polynomials are proportional to the elementary symmetric functions. For the case p 12 this leads to evaluations of these sums. Citeseerx krawtchouk polynomials and krawtchouk matrices.
Krawtchouk polynomials, proved by eagleson 1969, with an alternative proof in diaconis and gri ths 2012, is that without loss of generality p 12 there is a probability distribution. The simplest case is the krawtchouk polynomials where the representation is explained in section2. We can group integral zeros of the krawtchouk polynomials into three classes. Krawtchouk polynomials and universal bounds for codes and designs in hamming spaces abstract. Swarttouw 2010, 14 give a detailed list of their properties. Multivariate krawtchouk polynomials play an increasingly important role, in mathematical physics as well as probability and statistics 4, 11. Download fulltext pdf on a family of 2variable orthogonal krawtchouk polynomials article pdf available in symmetry integrability and geometry methods and applications 6 december 2010 with. Multivariate krawtchouk polynomials and composition birth. After this introduction, we continue with the probabilistic construction of krawtchouk.
Khare and zhou 27 use several systems of reproducing kernel polynomials, including those on the multinomial, in markov chain rate of convergence problems. Coding theory lecture notes nathan kaplan and members of the tutorial september 7, 2011 these are the notes for the 2011 summer tutorial on coding theory. Krawtchouk polynomials, the lie algebra sl 2, and leonard pairs article pdf available in linear algebra and its applications 4371. Then the set odp d z iz w, di, called the orbit ofd.
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