Related papers: Sheffer Polynomials and the s-ordering of Exponent…

Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time…

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them, its explicit expression, the connection with the Euler numbers, its integral representation via the Kontorovich-Lebedev…

Let $S=\{p_1,\dots,p_s\}$ be a finite non-empty set of distinct prime numbers, let $f\in \mathbb{Z}[X]$ be a polynomial of degree $n\ge 1$, and let $S'\subseteq S$ be the subset of all $p\in S$ such that $f$ has a root in $\mathbb{Z}_p$.…

In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study…

A coherent state representation of the expectation value of an arbitrary (but still polynomial) normal ordered quantum operator is discussed. This serves as a basis for developing a fast and easy-to-handle algorithm, based on series of…

The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\chi$-spherical functions indexed by the integers for each Hermitian quantum symmetric pair. A distinguished family of…

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr…

This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of…

The problem of quantizing a bivariate dynamical system can be reduced to evaluating the ordering of $\hat{q}^j \hat{p}^k$. Here, we consider the Weyl ordering of $\hat{q}^j \hat{p}^k$ that is then expressed in term of the annihilation…

We consider the Hopf algebra of B-diagrams as an algebra projecting onto the Heisenberg algebra and designed to encode the combinatorics of the bosonic normal-ordering problem. In order to understand and generalize the properties of the…

The exactly solvable eigenproblems in Schr\"odinger quantum mechanics typically involve the differential "shift operators". In the standard supersymmetric (SUSY) case, the shift operator turns out to be of first order. In this work, I…

In this paper we address the classical question going back to S. Bochner and H.L. Krall to describe all systems {p_{n}(x)} of orthogonal polynomials (OPS) which are the eigenfunctions of some finite order differential operator, i.e. satisfy…

We use the Poisson kernel of the continuous $q$-Hermite polynomials to introduces families of integral operators, which are semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The…

We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left…

We introduce a new algebraic construction, {\em monop}, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of…

We have previously observed an irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation, manifesting complex branching processes via the interplay of…

We introduce the theory of normal ordered grammars, which gives a natural generalization of the normal ordering problem. To illustrate the main idea, we explore normal ordered grammars associated with the Eulerian polynomials and the…

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…

We derive a normal ordering formula for the operator \((xI)^n\), where \(I\) denotes the Volterra operator. The resulting coefficients are shown to coincide with the Bessel numbers. We also present two applications, along with a…

We discuss two distinct aspects in supersymmetric quantum mechanics. First, we introduce a new class of operators A and $\bar{A}$ in terms of anticommutators between the momentum operator and N+1 arbitrary superpotentials. We show that…