Related papers: Coherent Orthogonal Polynomials

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions.…

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…

We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form $W^{(\nu)}_{\phi}(x) = x^{\nu}e^{-\phi(x)} W^{(\nu)}_{pol}(x)$, where $\nu>0$,…

Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…

By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for which $$ (ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\,\mathrm{I}$ is the identity operator, $x$…

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x){\rm d}x = \log \frac{2k}{1-x}{\rm d}x$ on $(-1,1)$, $k > 1$, and verify a conjecture of…

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…

Orthosymplectic Lie superalgebras are fundamental symmetries in modern physics, such as massive supergravity. However, their representations are far from being thoroughly understood. In the present paper, we completely determine the…

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $r,m\ge 2$. The universal central extension of the superelliptic current algebra $\mathfrak{g}\otimes A$ is $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A…

The central concept in the harmonic analysis of a compact group is the completeness of Peter-Weyl orthonormal basis as constructed from the matrix coefficients of a maximal set of irreducible unitary representations of the group, leading…

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb…

We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original functor but allows us to prove the…

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by…

We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of…

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and…

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is…

We study in this paper the infinite-dimensional orthogonal Lie algebra $\mathcal{O}_C$ which consists of all bounded linear operators $T$ on a separable, infinite-dimensional, complex Hilbert space $\mathcal{H}$ satisfying $CTC=-T^*$, where…

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n…

This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant…