Related papers: Macdonald operators at infinity
The quasisymmetic Macdonald polynomials $G_{\gamma}(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_{\lambda}(X; q, t)$ with the…
Kerov Hamiltonians are defined as a set of commuting operators which have Kerov functions as common eigenfunctions. In the particular case of Macdonald polynomials, well known are the exponential Ruijsenaars Hamiltonians, but the…
We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…
The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have…
We prove analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic "hypergeometric" sums as skew…
In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive…
Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…
In this paper, a link between $q$-difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable $x$ by $ q^{-n}$ in a Sturm-Liouville $q$-difference equation we discovered the Jacobi operator. With…
The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to…
We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under…
We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to…
We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…
The Hardy operator has all the monomial functions as eigenvectors. We study bounded operators on L^2 that take monomial functions to multiples of other monomials, with a shifted exponent. We prove that they all leave the space of functions…
In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck…
We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters…
By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is…
We indicate smooth real commuting matrix differential operators whose eigenvalues and eigenfunctions are parametrized by two-dimensional principally polarized abelian varieties.
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…
Inhomogeneous analogues of symmetric and nonsymmetric Macdonald polynomials were introduced by F. Knop and the author. In the symmetric case A. Okounkov has recently proved a beautiful expansion formula which can be viewed as a…
We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain…