Related papers: On the $q$-Dyson orthogonality problem
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 K\'{a}rolyi, Lascoux and Warnaar…
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…
In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson constant term identity. The non-zero part of Kadell's conjecture is a constant term identity indexed by a weak…
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…
In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald's constant term identities admit an extra set of free parameters, thereby…
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…
Using vertex operator we study Macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of Macdonald functions and thus give a generalized Frobenius…
Ramanujan provided several results involving the modified Bessel function $K_z(x)$ in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eiesenstien series on…
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.
We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously…
It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the $q$-Morris constant term identity. This conjecture was proved and…
We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence…
We give a Laurent series proof of the Habsieger-Kadell $q$-Morris identity, which is a common generalization of the $q$-Morris identity and the Aomoto constant term identity. The proof allows us to extend the theorem for some additional…
We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two quasi-periods. They are natural analogues of the…
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate…
We give a formal Laurent series proof of Andrews's $q$-Dyson Conjecture, first proved by Zeilberger and Bressoud.
We present an asymmetric $q$-Painlev\'e equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlev\'e equation (up to a simple…
It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the…
We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete…
We propose that geometric quantization of symplectic manifolds is the arrow part of a functor, whose object part is deformation quantization of Poisson manifolds. The `quantization commutes with reduction' conjecture of Guillemin and…