A Short Proof of the Zeilberger-Bressoud $q$-Dyson Theorem

A simple proof of the Zeilberger-Bressoud q-Dyson theorem

As an application of the Combinatorial Nullstellensatz, we give a short polynomial proof of the q-analogue of Dyson's conjecture formulated by Andrews and first proved by Zeilberger and Bressoud.

How to Extend Karolyi and Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem in order to Evaluate ANY Coefficient of the q-Dyson Product

We show how to extend the Karolyi-Nagy beautiful proof of the Zeilberger-Bressoud q-Dyson theorem, (first proved by Zeilberger and Bressoud in 1985, and originally conjectured by George Andrews in 1975), that states that the constant term of a certain Laurent polynomial equals the q-multinomial coefficient, how to evaluate any other specific coefficient. The algorithm implies that any such coefficient is always a certain rational function (that the algorithm finds) times the q-multinomial coefficient.

A symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Károlyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.

A Family of $q$-Dyson Style Constant Term Identities

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the $q$-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of $SL(n,\mathbb{C})$.

高斯多项式的新组合证明及 Garvan 的 $k$-秩的单调性

New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank

本文通过构造高斯多项式上的新双射，细化算法 Z，并给出广义 Rogers-Ramanujan 恒等式及 Garvan 的 $k$-秩单调性的组合证明。

A recursion for a symmetric function generalization of the $q$-Dyson constant term identity

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the $q$-Dyson constant term identity or the Zeilberger--Bressoud $q$-Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition $v=(v_1,\dots,v_n)$ in the case when only one $v_i\neq 0$. This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above mentioned constant term in the case when all the parts of $v$ are distinct. Recently we obtain a recursion for this…