Ordered products, W∞-algebra, and two-variable, definite-parity, orthogonal polynomials

Abstract

It has been shown that the Cartan subalgebra of W ∞ -algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. Also has been shown that they carry the infinite dimensional irreducible representation of the su(1, 1) algebra having the spectrum bounded from below. A realization of this algebra in terms of difference operators is also obtained. For particular values of the ordering parameter s they are identified with the classical orthogonal polynomials of a discrete variable, such as the Meixner, Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable s they satisfy a second order eigenvalue equation of hypergeometric type. Exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials. It has been put forward that it is the İnönü-Wigner contraction and its inverse that form bridge between the difference and differential calculus.

A connection between operator orderings and representations of the Lie algebra \mathfrak{sl}_2

“…It has previously been observed by several authors, cf e.g. [5,3,18] that some prescriptions for the operator orderings give rise to realizations of representations of the Lie algebra sl 2 acting on the space of quantum mechanical operators depending polynomially on p and q. Here, we extend these investigations somewhat further by establishing a close relationship between coefficients entering into a general ordering formula, cf (5.1) below, of a product pn qm and the covariance properties with respect to the transferred action of the Lie algebra sl 2 on the polynomial algebra of classical phase-space variables p and q.…”

“…Our interest in the problem was stimulated, on one hand, by the viewpoint of Howe, who explores the role that dual pairs play in the structure of the Weyl algebra [12,13], and on the other hand by the discovery of various connections of this problem with the special functions theory, among others done in the papers [2,3,5,15,16]. Verc ¸in in a recent paper [18] systematically investigated properties of a class of polynomials connected with a certain oneparameter family of orderings, while independently although somewhat later [10] that class of polynomials was investigated from the viewpoint exposed in the present paper by Gnatowska in her Doctor's Thesis at the University of Warsaw-cf also [11].…”

Relative Fisher information of discrete classical orthogonal polynomials

“…It governs the behaviour of the ratio P n+1 (x)/P n (x) when n goes to infinity as Rakhmanov showed in 1977 [44], and it characterizes the quantummechanical probability densities of ground and excited states of numerous physical systems (see, e.g. [17,40,41,52]). Beyond the variance, the Rényi and Shannon information entropies have been used to quantify the spreading of various simple discrete distributions.…”

“…The knowledge of the algebraic and spectral properties of the orthogonal polynomials in a discrete variable [1,13,28,32,40] as well as the elucidation of their universal structure [42] have been issues of permanent interest since the early years of the last century up until now, not only because of its mathematical interest [3,4,6,13,24,27,28,32,34,36,40,42] but also because of the increasing number of applications of these functions in so many scientific and technological fields [9, 22-24, 35-39, 49, 52]. In particular, the classical or hypergeometric discrete orthogonal polynomials do not only play a relevant role in the theory of difference analogues of special functions and other branches of mathematics [2,10,22,24,36,40,49], but also for mathematical modelling of a great deal of simple [8,9,35,37,39,49,50,52] and complex [12,14,23,23,38,48] systems, as well as for the compression of information for signal processing [29,41,42].…”

Image Description With Nonseparable Two-Dimensional Charlier and Meixner Moments

“…14 and 15. Although bivariate orthogonal polynomials play an important role in applications such as numerical analysis, optics, and quantum mechanics, 11,12,18 much less attention has been paid to the application of bivariate orthogonal polynomials in image analysis.…”