Related papers: Ladder Operators for q-orthogonal Polynomials

We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…

We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as…

We demonstrate in some detail how Macdonald polynomials emerge from the recently introduced 3-Schur functions when the plane-partition vector time-variables are projected onto the ordinary scalar times under non-vanishing angles, which…

Many interesting families of polynomials are indexed by permutations or related objects, and are defined by applying divided difference operators, modified by polynomials, on some initial base case. The fact that these constructions produce…

The logarithm-determinant is an widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories,…

For operators generated by a certain class of infinite band matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order finite difference equations. This…

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…

Let $M$ be a connected smooth manifold, let $\operatorname{Aut}(p)$ be the group automorphisms of the bundle $p\colon \mathbb{R}\times M\to \mathbb{R}$, and let $q\colon J^1(\mathbb{R},M)\times \mathbb{R\to }J^1(\mathbb{R},M)$ be the…

We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator with measurable coefficients. Amongst other…

We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and…

We investigate on some Appel-type orthogonal polynomial sequences on q-quadratic lattices and we provide some entire new characterizations of the Al-Salam Chihara polynomials (including the Rogers q-Hermite polynomials). The corresponding…

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 an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted…

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…

We show that hierarchies of differential Schroedinger operators for identical particles which are separating for the usual (anti-)symmetric tensor product, are necessarily linear, and offer some speculations on the source of quantum…

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical…

By a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with…

An O-operator is a relative version of a Rota-Baxter operator and, in the Lie algebra context, is originated from the operator form of the classical Yang-Baxter equation. We generalize the well-known construction of dendriform dialgebras…

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…

We define a Lax operator as a monic pseudodifferential operator $L(\partial)$ of order $N\geq 1$, such that the Lax equations $\dfrac{\partial L(\partial)}{\partial t_k}=[(L^{\frac kN}(\partial))_+,L(\partial)]$ are consistent and non-zero…