Related papers: Signal processing, orthogonal polynomials, and Heu…
This survey, aimed at information processing researchers, highlights intriguing but lesser known results, corrects misconceptions, and suggests research areas. Themes include: certainty in quantum algorithms; the "fewer worlds" theory of…
We derive five classes of quantum time-dependent two-state models solvable in terms of the double confluent Heun functions, five other classes solvable in terms of the biconfluent Heun functions, and a class solvable in terms of the…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
Given a sequence of polynomials $(p_n)_n$, an algebra of operators $\mathcal A$ acting in the linear space of polynomials and an operator $D_p\in \mathcal A$ with $D_p(p_n)=\theta_np_n$, where $\theta_n$ is any arbitrary eigenvalue, we…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
In this rather popular paper we stress some basics of the theory and the huge amount of applications of the Heun functions, as well as the still existing basic unsolved problems. The point is to attract more attention to this modern…
The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this…
Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy…
Integral relations and transformation rules are used to obtain, out of an asymptotic solution, a new group of four pairs of solutions to the double-confluent Heun equation. Each pair presents the same series coefficients but has solutions…
Some Open Problems Concerning Orthogonal Polynomials.
Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the…
This paper gives an introduction to the theory of orthogonal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals, and Spherical Harmonic series for…
In this note devoted to some aspects of the inverse problem of representation theory the attention is concentrated on the interrelations between various algebraic structures (algebras with operators) unraveled by different solutions of the…
In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on…
Some algebraic, geometric and geometroalgebraic characteristics of pairs of operators are discussed.
Explicit solutions for the three-term recurrence satisfied by associated continuous dual $q$-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The…
A class of non-linear eigenvalue problems defined in the form of operator polynomials is investigated. The problems are related to wave equations which appear in a relativistic quantum field theory. Spectral asymptotics for this class are…
Using the concept of $\mathcal{D}$-operator and the classical discrete family of dual Hahn, we construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of higher order difference operators.