Báo cáo toán học: "A New Approach to the Dyson Coefficients"

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