TAILIEUCHUNG - Báo cáo toán học: "On Kadell’s two Conjectures for the q-Dyson Product"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On Kadell’s two Conjectures for the q-Dyson Product. | On Kadell s two Conjectures for the q-Dyson Product Yue Zhou School of Mathematical Science and Computing Technology Central South University Changsha 410075 . China nkzhouyue@ Submitted Sep 7 2010 Accepted Dec 26 2010 Published Jan 2 2011 Mathematics Subject Classifications 05A30 33D70 Abstract By extending Lv-Xin-Zhou s first layer formulas of the q-Dyson product we prove Kadell s conjecture for the Dyson product and show the error of his q-analogous conjecture. With the extended formulas we establish a q-analog of Kadell s conjecture for the Dyson product. 1 Introduction In 1962 Freeman Dyson 3 conjectured the following constant term identity. Theorem Dyson s Conjecture . For nonnegative integers a0 a1 . an CT n 1 - xi T a L. . x Xj Ị ao a1 an 0 i j n v 3 01 n where a a0 a1 an and CTx f x means to take constant term in the x s of the series f x . The conjecture was quickly proved independently by Gunson 6 and by Wilson 15 . An elegant recursive proof was published by Good 5 and a combinatorial proof was given by Zeilberger 16 . In 1975 George Andrews 1 came up with a q-analog of the Dyson conjecture. Theorem . Zeilberger-Bressoud . For nonnegative integers a0 a1 . an q a qK qk q an where z m 1 z 1 zq 1 zqm 1 . THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P2 1 The Laurent polynomials in the above two theorems are respectively called the Dyson product and the q-Dyson product and denoted by Dn x a and Dn x a q respectively where x x0 . xn and a a0 . an . The Zeilberger-Bressoud q-Dyson Theorem was first proved combinatorially by Zeil-berger and Bressoud 17 in 1985. Recently Gessel and Xin 4 gave a very different proof by using the properties of formal Laurent series and of polynomials. The coefficients of the Dyson and the q-Dyson product were researched in 2 7 8 9 11 12 13 . In the equal parameter case the identity reduces to Macdonald s constant term conjecture 10 for root systems of type A. In 1988 Stembridge 14 gave the first layer .

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