The askey scheme for hypergeometric orthogonal polynomials. Although the basic properties of these polynomials are to some extent known, there is, to my knowledge, no detailed development available. Theset obis compactin the pointwise topologyfromh,andhence in theweaktopologyas asubset ofch. Krawtchouk polynomials appear in a variety of contexts. Ieee proof liu et al fractional krawtchouk transform with an application to image watermarking 3 theorem 1 speci. The zeros of generalized krawtchouk polynomials are studied. Koornwinder, krawtchouk polynomials, a unification of two different group theoretic interpretations siam j. 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.
The krawtchouk polynomials are proportional to the elementary symmetric functions. Integral zeroes of krawtchouk polynomials by ahmadm. A simple model for meixnerpollaczek, krawtchouk and meixner polynomials is introduced in section 2 and four more limit formulas fig. A generating function for the symmetric functions on the right of. 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. Krawtchouk polynomials and universal bounds for codes and designs in hamming spaces abstract. A uniform asymptotic expansion for krawtchouk polynomials. Krawtchouk polynomials encyclopedia of mathematics. In this paper, we stress this viewpoint by giving a new, uniform proof that the qkrawtchouk polynomials are the eigenvalues of these association schemes. For general properties of roots of krawtchouk polynomials see 26, 40. A problem in potential theory and zero asymptotics of. 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. This method is used to form a measure algebra whose characters are krawtchouk polynomials.
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. Survey of binary krawtchouk polynomials request pdf. Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. The algebraic interpretation is used to derive the main properties of the polynomials. The integral zeros for two families of qkrawtchouk polynomials are classi. Then the set odp d z iz w, di, called the orbit ofd. In mathematics, the dual qkrawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic askey scheme. They form an important family of orthogonal polynomials 5, 16, 23.
Sums of squares of krawtchouk polynomials, catalan. Reconstruction problems for graphs, krawtchouk polynomials. A short proof of a generating function for jacobi polynomials, proc. Multivariate krawtchouk polynomials and a spectral theorem. Departmentofmathematics bruneluniversity uk october,2012 1. The krawtchouk and multivariate krawtchouk polynomials are symmetric functions of orthogonal functions sets on each of the trials. Thanks for contributing an answer to mathematics stack exchange. The multivariate krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states. Next, 1 we construct a set of orthonormal eigenvectors of matrix k 2 corresponding to the two eigenvalues. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Attention is drawn to a phenomenon ofpseudostabilityin connection with the threeterm recurrence relation for discrete orthogonal polynomials. Krawtchouk polynomials, a unification of two different. A detailed study of integral roots of binary krawtchouk polynomials was undertaken in 9, 16. Ieee transactions on signal processing 1 fractional.
Retrieved from the university of minnesota digital. We would like to mention several questions which appear to be out of the scope of the paper but very much similar to its. Since the 41 weighted krawtchouk polynomials are discrete, there is no nu42 merical approximation in deriving the transform coef. We show how they arise as condensed sylvesterhadamard matrices via a binary shuffling function. The nonsymmetric krawtchouk polynomials have received by the editors june 3, 1996 and, in revised form, november 1, 1998. Multivariate krawtchouk polynomials and composition birth and. The multivariate krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of n multinomial trials. Multivariate krawtchouk polynomials play an increasingly important role, in mathematical physics as well as probability and statistics 4, 11.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Krawtchouk polynomials appear originally as orthogonal polynomials for the. For additional properties of these polynomials, see the references cited in 4, p. Kravchuk polynomials article about kravchuk polynomials by. Krawtchouk polynomials and the symmetrization of hypergroups. The underlying symmetric tensor algebra is then presented. Krawtchouk polynomials and krawtchouk matrices arxiv.
The proof uses several recurrences and other properties satisfied by the krawtchouk polynomials 17. Is the recurrence relation for orthogonal polynomials. Hecke algebra, qkrawtchouk polynomial, zonal spherical function. Citeseerx krawtchouk polynomials and krawtchouk matrices. The columns of krawtchouk matrices may be considered generalized binomial coe. Let wbe a compact group ofautomorphisms on the compact p,hypergroup h. Hecke algebra, q krawtchouk polynomial, zonal spherical function.
Dunkls interpretation of krawtchouk polynomials as spherical functions on wreath products of symmetric groups is generalized to the case of intertwining functions. For the case p 12 this leads to evaluations of these sums. The dual multivariate krawtchouk polynomials, which also have a polynomial structure, are seen to occur. The integral zeros for two families of q krawtchouk polynomials are classi. Generalizing krawtchouk polynomials using hadamard matrices. The paper begins with expressing the 1dimensional krawtchouk polynomials as symmetric functions of bernoulli trials. We first give some basic properties of the qkrawtchouk polynomials, without proof. Multivariate krawtchouk polynomials and composition birth.
Writing x, y for x0,x1, we have in degree n for the kth. Hahn polynomials and multivariate krawtchouk polynomials 16,17,20,35. The result follows from theorem 1 and the binomial theorem. In this context, tichy and the author recently showed that a binary diophantine. Krawtchouks polynomial for hypergeometric distribution. We can group integral zeros of the krawtchouk polynomials into three classes. Krawtchouk polynomials, catalan numbers, super catalan numbers, boolean lattice, algebras, zeons. The approach of 21 on composition markov chains ties in with the multivariate polynomials as well, cf. 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. For instance, in 12, sharapudinov has obtained the asymptotic formula.
Sums of squares of krawtchouk polynomials, catalan numbers. The simplest case is the krawtchouk polynomials where the representation is explained in section2. 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. Journal of statistical planning and inference 154, 3953.
Krawtchouk orthogonal polynomials the krawtchouk orthogonal polynomials knx. Pdf krawtchouk matrices have as entries values of the krawtchouk polynomials for nonnegative integer arguments. Biorthogonal codes with spherically restricted positions. When q 2, these polynomials are called binary krawtchouk polynomials. In applications the most important case is the binary krawtchouk polynomials corresponding to q 1 2. Krawtchouk polynomials and universal bounds for codes and.
Krawtchouk matrices have as entries values of the krawtchouk polynomials for nonnegative integer arguments. This hypergroup property leads to a characterization of reversible markov chains with multinomial stationary distributions and multivariate krawtchouk eigenfunctions. On characterizing integral zeros of krawtchouk polynomials by. Thus the krawtchouk polynomials will be considered to be the special case. Is the recurrence relation for orthogonal polynomials always. The multivariate krawtchouk polynomials are extensions of this construction in higher dimensions. Integer zeros of qkrawtchouk polynomials in classical. Pdf orthogonality relations for multivariate krawtchouk. Theqkrawtchouk polynomials are the spherical functions for three different chevalley groups over a finite field. Krawtchouk and multivariate krawtchouk polynomial hypergroups. The necessary and sufficient conditions for attainability of the bounds are found. Recently, there is a considerable interest in the asymptotics of krawtchouk polynomials, when the degree n grows to infinity. On characterizing integral zeros of krawtchouk polynomials. Saff2 department of mathematics, institute for constructive mathematics, university of south florida, tampa, florida 33620, u.
A problem in potential theory and zero asymptotics of krawtchouk polynomials p. Some qkrawtchouk polynomials on chevalley groups, amer. In the multivariate krawtchouk polynomials, there is not a. Kravchuk polynomials or krawtchouk polynomials also written using several other transliterations of the ukrainian name. 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. Theq krawtchouk polynomials are the spherical functions for three different chevalley groups over a finite field. Reproducing kernel orthogonal polynomials on the multinomial. Orthogonality relations for multivariate krawtchouk polynomials. Khare and zhou 27 use several systems of reproducing kernel polynomials, including those on the multinomial, in markov chain rate of convergence problems. In this case, the series is called a hypergeometric polynomial. Dragnev1 department of mathematics, indianapurdue university, fort wayne, indiana 46805, u. Using techniques of dunkl to decompose the irreducible representations with respect to a maximal parabolic subgroup, we derive three addition theorems. We begin with krawtchouk polynomials for general parameter p and derive an identity for partial sums of squares along a column. The multivariate quantum qkrawtchouk polynomials are shown to arise as matrix elements of qrotations acting on the state vectors of many qoscillators.
In recent years, the study of the asymptotic property for krawtchouk polynomials appears continuously in some important mathematics publications. On zeros of discrete orthogonal polynomials sciencedirect. 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. Uvarov, special functions of mathematical physics, birkhauser 1988 translated from russian. A partially ordered set and gkrawtchouk polynomials. Krawtchouk polynomials, the lie algebra sl 2, and leonard pairs article pdf available in linear algebra and its applications 4371.
A conceptual unification is given of these two group theoretic interpretations of krawtchouk polynomials. Swarttouw 2010, 14 give a detailed list of their properties. In section 1, we find a model for hahn polynomials and prove the seven arrows of fig. They are symmetric with respect to n 2, so x k n 2. The associated polynomials are related to affine matrix groups. N vn k nj,n the polynomials knj,n are orthogonal with respect to the binomial distribution with parameters n,p. On integral zeros of krawtchouk polynomials semantic scholar. Pdf krawtchouk polynomials and krawtchouk matrices. 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. Pdf on a family of 2variable orthogonal krawtchouk. Pdf krawtchouk polynomials, the lie algebra sl 2, and. An algebraic interpretation of the multivariate q krawtchouk. But avoid asking for help, clarification, or responding to other answers. Research article generalizing krawtchouk polynomials using.
After this introduction, we continue with the probabilistic construction of krawtchouk. 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. Research article generalizing krawtchouk polynomials using hadamard matrices peters. A selfcontained proof of optimality of these bounds in the framework of the linear programming method is given.
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