Related papers: Super-Macdonald polynomials: Orthogonality and Hil…
Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our…
The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical…
We give a complete classification in canonical forms on finite-dimensional vector spaces over the real numbers.
This thesis is divided into three parts. The first part deals with cylindric plane partitions. The second with lambda-determinants and the third with commutators in semi-circular systems. For more detailed abstract please see inside.…
Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra…
In this paper, we introduce the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szeg\"o polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szeg\"o…
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…
In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality…
We introduce new classes of right quaternionic Hilbert spaces of Bargmann-Fock type $\mathcal{GB}_{m}^{2}(\mathbb{H})$, labeled by nonnegative integer $m$, generalizing the so-called slice hyperholomorphic Bargmann-Fock space introduced…
We construct the generalized $\beta$ and $(q,t)$-deformed partition functions through $W$ representations, where the expansions are respectively with respect to the generalized Jack and Macdonald polynomials labeled by $N$-tuple of Young…
We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…
We give an explicit formula for an operator that sends a wreath Macdonald polynomial to the delta function at a character associated to its partition. This allows us to prove many new results for wreath Macdonald polynomials, especially…
Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green…
Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are…
We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter…
Koornwinder polynomials are $q$-orthogonal polynomials equipped with extra five parameters and the $B C_n$-type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariate analogue of Askey-Wilson polynomials. They are…
Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e…
For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a…
We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing…
We establish orthogonality relations for the Baker-Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik-Macdonald-Mehta integral for these functions. As a corollary, we give a simple…