scholarly journals Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

2020 ◽  
pp. 2140016
Author(s):  
W. Riley Casper ◽  
Stefan Kolb ◽  
Milen Yakimov
Keyword(s):  
Recurrence Relations ◽  
Star Product ◽  
Hermite Polynomials ◽  
Star Products ◽  
Defining Relations ◽  
Algebraic Properties ◽  
Product Method ◽  
Usual Quantum ◽  
Multivariate Orthogonal Polynomials ◽  
Nichols Algebras

We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

scholarly journals Defining relations of quantum symmetric pair coideal subalgebras

2021 ◽  
Vol 9 ◽  
Author(s):  
Stefan Kolb ◽  
Milen Yakimov
Keyword(s):  
Chebyshev Polynomials ◽  
Hermite Polynomials ◽  
Symmetric Pair ◽  
Star Products ◽  
Graded Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
New Family

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$ -graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

scholarly journals Recurrence relations for exceptional Hermite polynomials

2016 ◽  
Vol 204 ◽  
pp. 1-16 ◽  
Author(s):  
D. Gómez-Ullate ◽  
A. Kasman ◽  
A.B.J. Kuijlaars ◽  
R. Milson
Keyword(s):  
Recurrence Relations ◽  
Hermite Polynomials

scholarly journals Hermitian realizations of κ-Minkowski space–time

2015 ◽  
Vol 30 (03) ◽  
pp. 1550019 ◽  
Author(s):  
Domagoj Kovačević ◽  
Stjepan Meljanac ◽  
Andjelo Samsarov ◽  
Zoran Škoda
Keyword(s):  
Minkowski Space ◽  
Star Product ◽  
Plane Waves ◽  
Space Time ◽  
Star Products ◽  
Systematic Construction ◽  
Minkowski Space Time ◽  
Noncommutative Plane

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

scholarly journals SCALE TRANSFORMATIONS ON THE NONCOMMUTATIVE PLANE AND THE SEIBERG–WITTEN MAP

2005 ◽  
Vol 20 (25) ◽  
pp. 5871-5890 ◽  
Author(s):  
A. PINZUL ◽  
A. STERN
Keyword(s):  
Star Product ◽  
Field Theories ◽  
Gauge Fields ◽  
Gauge Transformations ◽  
Commutation Relations ◽  
Defining Relations ◽  
Scale Transformations ◽  
Noncommutative Plane ◽  
Noncommutative Parameter

We write down three kinds of scale transformations (i)–(iii) on the noncommutative plane. Transformation (i) is the analogue of standard dilations on the plane, transformation (ii) is a rescaling of the noncommutative parameter θ, and transformation (iii) is a combination of the previous two, whereby the defining relations for the noncommutative plane are preserved. The action of the three transformations is defined on gauge fields evaluated at fixed coordinates and θ. The transformations are obtained only up to terms which transform covariantly under gauge transformations. We give possible constraints on these terms. We show how the transformations (i) and (ii) depend on the choice of star product, and show the relation of (ii) to Seiberg–Witten transformations. Because transformation (iii) preserves the fundamental commutation relations it is a symmetry of the algebra. One has the possibility of implementing it as a symmetry of the dynamics, as well, in noncommutative field theories where θ is not fixed.

MOYAL STAR PRODUCT OF μ-HOLOMORPHIC j-DIFFERENTIALS

2008 ◽  
Vol 05 (03) ◽  
pp. 363-373
Author(s):  
M. KACHKACHI
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Quantum Effect ◽  
Star Product ◽  
Complex Structure ◽  
Complex Structures ◽  
Star Products ◽  
First Order ◽  
Conformal Fields ◽  
Conformal Covariance

It was shown in [1], only for scalar conformal fields, that the Moyal–Weyl star product can introduce the quantum effect as the phase factor to the ordinary product. In this paper we show that, even on the same complex structure, the Moyal–Weyl star product of two j-differentials (conformal fields of weights (j, 0)) does not vanish but it generates the quantum effect at the first order of its perturbative series. More generally, we get the explicit expression of the Moyal–Weyl star product of j-differentials defined on any complex structure of a bi-dimensional Riemann surface Σ. We show that the star product of two j-differentials is not a j-differential and does not preserve the conformal covariance character. This can shed some light on the Moyal–Weyl deformation quantization procedure connection's with the deformation of complex structures on a Riemann surface. Hence, the situation might relate the star products to the Moduli and Teichmüller spaces of Riemann surfaces.

scholarly journals The Hermite polynomials and the Bessel functions from a general point of view

2003 ◽  
Vol 2003 (57) ◽  
pp. 3633-3642 ◽  
Author(s):  
G. Dattoli ◽  
H. M. Srivastava ◽  
D. Sacchetti
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View ◽  
General Point ◽  
Ordinary Case

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

scholarly journals Comparison and continuity of Wick-type star products on certain coadjoint orbits

2019 ◽  
Vol 31 (5) ◽  
pp. 1203-1223
Author(s):  
Chiara Esposito ◽  
Philipp Schmitt ◽  
Stefan Waldmann
Keyword(s):  
Star Product ◽  
Coadjoint Orbit ◽  
Type Star ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Continuity Properties ◽  
Fréchet Algebra

AbstractIn this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

scholarly journals ON RELATION BETWEEN WEYL AND KONTSEVICH QUANTUM PRODUCTS: DIRECT EVALUATION UP TO THE ℏ3-ORDER

2001 ◽  
Vol 16 (10) ◽  
pp. 615-625 ◽  
Author(s):  
A. ZOTOV
Keyword(s):  
Deformation Quantization ◽  
Jacobi Identity ◽  
Star Product ◽  
Star Products ◽  
Moyal Product ◽  
Third Order ◽  
Direct Evaluation ◽  
The Third ◽  
Arbitrary Change ◽  
Poisson Bivector

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Weyl (Moyal) product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bivector is shown to depend on ℏ and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

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