Related papers: Division of Differential operators, intertwine rel…
How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Fa\'a di Bruno formula with Bell polynomials; while there are extensions of the Fa\'a di Bruno formula,…
Dunkl operators are differential-difference operators on $\b R^N$ which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
We develop the algebraic approach to duality, more precisely to intertwinings, within the context of particle systems in general spaces, focusing on the $\mathfrak{su}(1,1)$ current algebra. We introduce raising, lowering, and neutral…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative…
Let $k \geq 2$ be an integer. We prove that factorization of integers into $k$ parts follows the Dirichlet distribution $\text{Dir}\left(\frac{1}{k},\ldots,\frac{1}{k}\right)$ by multidimensional contour integration, thereby generalizing…
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 develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level…
In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
Let D_v the difference operator and q-difference operators defined by D_\omega p(x) = \frac{p(x+\omega)-p(x)}{\omega} and D_q p(x) = \frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let…
Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…
We analyze a certain class of integral equations related to Marchenko equations and Gel'fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank…
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for…
We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n) T^{-1} where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We…
The main objective of this work is to study generalized Browder's and Weyl's theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $\tau_{A,B}=L_AR_B$.
Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form.…
Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…