Related papers: Gauss hypergeometric function and quadratic $R$-ma…
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order…
In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators. We also realize them in terms of…
Huang's geometric interpretation of vertex operator algebras is extended to a supergeometric interpretation of vertex operator superalgebras. In particular, the geometry of spheres with punctures and local analytic coordinates in terms of…
We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to…
We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover,…
The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R-operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded…
The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratic-type lattices. We…
New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…
We study the action of the Hecke operators Un on the set of hy- pergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set n^a with a an integer…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
In this paper we study self-adjoint commuting ordinary differential operators with polynomial coefficients. These operators define commutative subalgebras of the first Weyl algebra. We find new examples of commuting operators of rank 2.
Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an…
Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in…
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. - A…
In this paper, we study scalar the forth order linear differential operators over an oriented 2-dimensional manifold. We investigate differential invariants of these operators and show their application to the equivalence problem.
For linear operators which factor with suitable assumptions concerning commutativity of the factors, we introduce several notions of a decomposition. When any of these hold then questions of null space and range are subordinated to the same…
The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary…
For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining…
In this paper, we give a new series of coboundary operators of Hom-Lie algebras. And prove that cohomology groups with respect to coboundary operators are isomorphic. Then, we revisit representations of Hom-Lie algebras, and generalize the…
In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group.Upon…