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:A New Approach to the Dyson Coefficients. | A New Approach to the Dyson Coefficients Sabrina . Pang College of Mathematics and Statistics Hebei University of Economics and Business Shijiazhuang 050061 . China stpangxingmei@ Lun Lv School of Science Hebei University of Science and Technology Shijiazhuang 050018 . China klunlv@ Submitted Feb 16 2010 Accepted Aug 16 2010 Published Aug 24 2010 Mathematics Subject Classifications 05A30 33D70 Abstract In this paper we introduce a direct method to evaluate the Dyson coefficients. 1 Introduction In 1962 Dyson 2 conjectured the following constant term identity. Theorem Dyson s Conjecture . For nonnegative integers a1 a2 . an CT Dn x a ai a2 an ai a2 an where CTx f x denotes the constant term and Dn x a . n 1 - yt- 1 j j O - j Dyson product Corresponding author THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N30 1 Dyson s conjecture was proved independently by Gunson 5 and Wilson 11 . In 1970 a brief and elegant proof was published by Good 4 . Later Zeilberger 13 gave a combinatorial proof. The q-analog of Theorem was conjectured by Andrews 1 in 1975 and was first proved combinatorially by Zeilberger and Bressoud 14 . Recently Gessel and Xin 3 gave a different proof by using properties of formal Laurent series. In recent years there has been increasing interest in evaluating the coefficients of monomials M nn i xbi where n i bi 0 in the Dyson product. Based on Good s proof Kadell 6 gave three non-constant term coefficients. Sills and Zeilberger 10 described an algorithm that automatically conjectures and proves closed-form expressions. Later Sills 9 extended Good s idea and obtained the closed-form expressions for M being Xs XỉX xxs respectively. By virtue of Zeilberger and Sills Maple package GoodDyson Lv Xin and Zhou 7 found two closed-form expressions for M that has a square in the numerator. Moreover by generalizing Gessel-Xin s method 3 for proving the Zeilberger-Bressoud q-Dyson Theorem Lv Xin and Zhou 8 established a family
