Sách A New Approach to the Dyson Coefficients

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 1.1 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 : =
bi = 0, in the Dyson product. Based on Good’s
proof, Kadell [6] gave three non-constant term coefficients. Sills and Zeilberger [10] de-
scribed 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
, 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 of q-Dyson style
constant term identities.