The Research Of Dyson Conjecture And Its Related Content

In 1962,Dyson speculated based on the question in particle physics that a constant term identity,or Dyson conjecture,is going to be mentioned below.Gunson and Wilson?Nobel prize winner?have proved the Dyson's conjecture.Shortly thereafter,Good applied Lagrange interpolation to give this conjecture a beautiful recursive proof.In addition,Zeilberger gives a combination of proof of this conjecture by using the Vandermonde determinant and multiple competition graphs.Many mathematicians have studied the Good proof of Dyson's conjecture and got a lot of results.The most notable is to calculate the coefficients of several Dyson products.Kadell sums up how to calculate the Dyson coefficients of M=x1/x2,M=?x₁×x₂?/(x_n-1×x_n and M=?x₁×x₂?/x_n² with these terms according to Good's thoughts,Zeilberger and Sills follow this idea to find an automatic guess and prove Dyson coefficient algorithm,So applying this algorithm,Sills deduced the Dyson coefficients of M=?x_s?/?x_t?,M=?x_s×x_t?/x_r² and M=?x₁×x_n?/?x₁×x_n?.Recently,Lv Lun,Xin Guo-ce and Zhou Yue obtained the Dyson coefficient without the square term in the denominator by extending the Gessel-Xin proving q-Dyson conjecture.In this paper,we discuss the Dyson constant terms identity.The main contributions are as follows:First,this paper gives the first guessing method of Dyson coefficient,that is,the initial condition is verified first,then the boundary condition is verified and the regression relation is finally verified,Furthermore,we also form a method of directly calculating the above Dyson coefficients,the advantage of this method is that the specific coefficients can be calculated directly without guess coefficients,this result naturally continues the work of the Dyson coefficients that do not contain square terms in the denominator of Zeilberger and Sills.Using the above results,we also construct several identities in addition to several Dyson type constant terms.Finally,this paper presents a new method to prove Dyson's conjecture.This method turns Dyson into a polynomial of quantitative,and gives the polynomial proof of Dyson conjecture from the equal of polynomials.