Related papers: A Rodrigues formula for the Jack polynomials and t…
By building a second order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second order difference equation of hypergeometric type on nonuniform lattices…
We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space $R^m$. In both cases we obtain several properties of these polynomials, such as a…
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald…
We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model's Hamiltonian on…
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…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered
We introduce a new operator $\Gamma$ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula provides a connection between the theory of Macdonald operators initiated by Bergeron,…
The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of…
Using the ground state $\psi_0$ of a multicomponent generalization of the Calogero-Sutherland model as a weight function, orthogonal polynomials in the coordinates of one of the species are constructed. Using evidence from exact analytic…
We give an explicit Pieri formula for Macdonald polynomials attached to the root system C_n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.
The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a…
We analyze a possible minimal counterexample to the Jacobian Conjecture $P,Q$ with $\gcd(deg(P),deg(Q))=16$ and show that its existence depends only on the existence of solutions for a certain Abel differential equation of the second kind.
We consider products of two Macdonald polynomials of type A, indexed by dominant weights which are respectively a multiple of the first fundamental weight and a weight having zero component on the k-th fundamental weight. We give the…
We introduce a new class of polynomials of multiple orthogonality with respect to the product of $r$ classical discrete weights on integer lattices with noninteger shifts. We give explicit representations in the form of the Rodrigues…
We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas…
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of…
In this work it is propose an alterative proof of one of basic properties of the zonal polynomials. This identity is generalised for the Jack polynomials.
Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this…
We consider Jack polynomials $J_\lambda$ and their shifted analogue $J^#_\lambda$. In 1989, Stanley conjectured that $\langle J_\mu J_\nu, J_\lambda \rangle$ is a polynomial with nonnegative coefficients in the parameter $\alpha$. In this…