Related papers: Gauss hypergeometric function and quadratic $R$-ma…
Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…
The paper is an investigation of the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These…
Theory of differential operators on associative algebras is not extended to the non-associative ones in a straightforward way. We consider differential operators on Lie algebras. A key point is that multiplication in a Lie algebra is its…
We introduce Wirtinger operators for functions of several quaternionic variables. These operators are real linear partial differential operators which behave well on quaternionic polynomials, with properties analogous to the ones satisfied…
In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this…
Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we…
We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…
We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary…
We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on non-homogeneous lattice, and extend these operators to the generalized orthogonal…
The aim of this paper is to show some examples of matrix-valued orthogonal functions on the real line which are simultaneously eigenfunctions of a second-order differential operator of Schr\"{o}dinger type and an integral operator of…
A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…
The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub--Welsh algorithm to…
The problem of equivalency for linear differential operators of the first order is discussed.
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape invariant operators. These operators can…
The generators of the algebra $gl_{n+1}$ in a form of differential operators of the first order acting on ${\bf R}^n$ with matrix coefficients are explicitly written. The algebraic Hamiltonians for a matrix generalization of $3-$body…
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 two examples of algebras of differential operators associated to families of matrix valued orthogonal polynomials arising from representations of SU$(N+1)$. The first one gives a commutative algebra and the second one a…
This paper deals with well-known higher-order generalizations of Hankel operators. We show that higher-order Hankel operators can be written explicitly as linear differential operators, and give the exact form of these differential…