Related papers: Krawtchouk matrices from the Feynman path integral…
Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic…
An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the 3-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of…
The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is…
Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. Although the basic properties of these polynomials are to some extent known, there is, to my knowledge, no detailed…
Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor…
In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…
We put together the ingredients for an efficient operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provide an inherently discrete alternative to Fourier…
Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate…
Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for…
Discretizations of the Feynman-Kac path integral representation of the quantum mechanical density matrix are investigated. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional function…
In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
The scattering properties of quantum particles on fractal potentials at different stages of fractal growth are obtained by means of the transfer matrix method. This approach can be easily adopted for project assignments in introductory…
In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we…
Functions of several quaternion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the $\tilde \partial $-equations are studied. Moreover, quaternion Stein manifolds are…
We present a new formula for the coaction of a large class of integrals. When applied to one-loop (cut) Feynman integrals, it can be given a diagrammatic representation purely in terms of pinches and cuts of the edges of the graph. The…
The purpose of this expository paper is to highlight the starring role of time-frequency analysis techniques in some recent contributions concerning the mathematical theory of Feynman path integrals. We hope to draw the interest of…
Krylov space methods provide an efficient framework for analyzing the dynamical aspects of quantum systems, with tridiagonal matrices playing a key role. Despite their importance, the behavior of such matrices from chaotic to integrable…
We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element…
We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a new functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As…