This survey article, written for the Encyclopedia of Mathematical Physics, 2nd edition, is
devoted to the remarkable family of operators introduced by Charles Dunkl and to their $ q …

We present four infinite families of mutually commuting difference operators which include
the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators …

We present how explicit eigenfunctions of the principal Hamiltonian for the BC m relativistic
Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and …

We establish a direct link between Dunkl operators and quantum Lax matrices LL for the
Calogero–Moser systems associated to an arbitrary Weyl group W (or an arbitrary finite …

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of
a finite reflection group W. We determine all such ideals that are invariant under the action of …

The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math
Phys 288: 653–675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) …

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic
hypergeometric series associated with A-type root systems of different ranks. By …

Inside the double affine Hecke algebra $\mathbb {H} _ {n, q,\tau} $ of type $ GL_n $, we
define a subalgebra $\mathbb {H}^{\mathfrak {gl} _n} $ that may be thought of as a $ q …

We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the
Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta …

For the class of quantum integrable models generated from the q− Onsager algebra, a basis
of bispectral multivariable q− orthogonal polynomials is exhibited. In the first part, it is shown …