Integer Zeros ofq-Krawtchouk Polynomials in Classical Combinatorics

Abstract

dedicated to dominique foata on the occasion of his 65th birthdayInteger zeros of binary Krawtchouk polynomials occur in various problems of classical combinatorics. We present some of these properties and generalise them to q-Krawtchouk polynomials. We also give a survey of what is known about these zeros.  2001 Elsevier Science

Generalizing Krawtchouk Polynomials Using Hadamard Matrices

“…Krawtchouk polynomials and their generalisation appear in many areas of mathematics, see 2 [8]: harmonic analysis [11,6,17], statistics [7], combinatorics and coding theory [5,9,10,12,14], probability theory [8], representation theory (e.g., of quantum groups) [1,3,13], difference equations [2], and pattern recognition [15]. However, our motivation in this article was primarily driven by generalising the results obtained in [16] -and thus shedding more light on the qualitative structure of (generalisations of) Krawtchouk polynomials -and not yet with a specific application in mind.…”

“…Other generalizations of binary Krawtchouk polynomials have also been considered. [9] generalized some properties of binary Krawtchouk polynomials to q-Krawtchouk polynomials. [1] derived orthogonality relations for quantum and q-Krawtchouk polynomials and showed that affine q-Krawtchouk polynomials are dual to quantum q-Krawtchouk polynomials.…”

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

“…for y ≈ Y − (z), p < z < 1 and unknown functions A (9) (z), B (9) (z). Using (87) in (97) we have, as β → ∞…”

“…Lloyd's theorem [23] states that if a perfect code exists in the Hamming metric, then the Krawtchouk polynomial must have integral zeros [3], [5], [20]. Not surprisingly, these zeros have been the subject of extensive research [4], [7] [9], [10], [11], [16], [35].…”

Μελέτη των ριζών των associated ορθογωνίων q-πολυωνύμων

“…Tèloc ta orjog¸nia q-polu¸numa kai oi rÐzec touc emfanÐzontai sthn sunduastik anlush kai sth jewrÐa kwdikopoÐhshc [11,18,43].…”