相关论文: Distributed Gaussian polynomials as q-oscillator e…

In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier…

Let $\mathrm{G} = \mathrm{Gl}_{n}(K)$, and $\mathrm{H} = \mathrm{G}^{\sigma}$ for $\sigma$ an involution of the form $g\rightarrow aga^{-1}$, It is known that for $K =\mathbb{Q}_q$ any irreducible representation of $\mathrm{G}$ with an…

The dynamical algebra of the q-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction…

$q,t$-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal $\mathfrak{gl}_1$ algebra. These representations are described in terms of finite difference equations…

We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally…

{Although q-oscillators have been used extensively for realization of quantum universal enveloping algebras,such realization do not exist for quantum matrix algebras ( deformation of the algebra of functions on the group ). In this paper we…

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…

We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical…

We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert…

We define two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Then, we proved that…

The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that…

Exact eigenvalue correlation functions are computed for large $N$ hermitian one-matrix models with eigenvalues distributed in two symmetric cuts. An asymptotic form for orthogonal polynomials for arbitrary polynomial potentials that support…

New nonlinear connection formulae of the q-orthogonal polynomials, such continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a…

We extend the approach to ${\tau}$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the…

This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal…

We construct path integral representations for the evolution operator of q-oscillators with root of unity values of q-parameter using Bargmann-Fock representations with commuting and non-commuting variables, the differential calculi being…

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail conjecture…

We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…

We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of…

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra…