Related papers: Macdonald operators at infinity
We show that the method of separation of variables gives a natural generalisation of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the ``quadratic'' integral…
We introduce certain raising and lowering operators for Macdonald polynomials (of type $A_{n-1}$) by means of Dunkl operators. The raising operators we discuss are a natural $q$-analogue of raising operators for Jack polynomials introduced…
In this paper we construct a discrete linear operator $K$ which transforms $A_2$ Macdonald polynomials into the product of two basic $3\phi_2$ hypergeometric series with known arguments. The action of the operator $K$ on power sums in two…
We construct explicitly non-polynomial eigenfunctions of the difference operators by Macdonald in case $t=q^k$, $k\in{\mathbb Z}$. This leads to a new, more elementary proof of several Macdonald conjectures, first proved by Cherednik. We…
Genus 2 Macdonald polynomials $\Psi^{(q,t)}_{j_1,j_2,j_3}$ generalize ordinary Macdonald polynomials in several aspects. First, they provide common eigenfunctions for commuting difference operators that generalize the Macdonald difference…
Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald…
We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.
We consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are…
Skewing operators play a central role in the symmetric function theory because of the importance of the product structure of the symmetric function space. The theory of noncommutative symmetric functions is a useful tool for studying…
We study a specific family of symmetric norms on the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space. With respect to each symmetric norm in this family the identity operator fails to attain…
In this paper we use orthonormal basis for the Hardy space $H^{2}(\mathbb{T})$, formed by rational functions, to characterize complex symmetric Toeplitz operators on $H^{2}(\mathbb{T})$. As a result, we get examples of these operators whose…
We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of raising/lowering operators with a simple action on…
Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of…
In this paper, we present a comprehensive account of all Laguerre-type differential operators $D$ that are symmetric with respect to a $2\times 2$ irreducible weight $W$ on the interval $(0, \infty)$. These operators are associated with…
The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $\theta_k$ are introduced, and the Hamiltonian is represented as a…
The $SL(2,\mathbb Z)$-symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a…
Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix…
We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the…
In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…