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Tom H. Koornwinder

Publications and source records attributed to Tom H. Koornwinder.

54 records · Page 3Linked to original sources
arXiv2007

The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

math.QA↗
arXiv2007

The structure relation for Askey-Wilson polynomials

An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n+1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.

math.CA↗
arXiv2006

LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas.

math.CA↗
arXiv2005

Lowering and raising operators for some special orthogonal polynomials

This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC_2 a new pair of lowering and raising operators is obtained.

math.CA↗
arXiv2003

A second addition formula for continuous q-ultraspherical polynomials

This paper provides the details of Remark 5.4 in the author's paper "Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group", SIAM J. Math. Anal. 24 (1993), 795-813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3phi2's as Fourier coefficients. The proof given there used the quantum group interpretation. Here this identity will be generalized to a 3-parameter class of Askey-Wilson polynomials being expanded in terms of continuous q-ultraspherical polynomials with a product of two 2phi2's as coefficients, and an analytic proof will be given for it. Then Gegenbauer's addition formula for ultraspherical polynomials and Rahman's addition formula for q-Bessel functions will be obtained as limit cases. This q-analogue of Gegenbauer's addition formula is quite different from the addition formula for continuous q-ultraspherical polynomials obtained by Rahman and Verma in 1986. Furthermore, the functions occurring as factors in the expansion coefficents will be interpreted as a special case of a system of biorthogonal rational functions with respect to the Askey-Roy q-beta measure. A degenerate case of this biorthogonality are Pastro's biorthogonal polynomials associated with the Stieltjes-Wigert polynomials.

math.CA↗
arXiv2000

Invariant differential operators on nonreductive homogeneous spaces

A systematic exposition is given of the theory of invariant differential operators on a not necessarily reductive homogeneous space. This exposition is modelled on Helgason's treatment of the general reductive case and the special non-reductive case of the space of horocycles. As a final application the differential operators on (not a priori reductive) isotropic pseudo-Riemannian spaces are characterized.

math.RT↗
arXiv2000

QDYBE: some explicit formulas for exchange matrix and related objects in case of sl(2), q=1

This mainly tutorial paper is intended as a somewhat larger example for parts of the theory exposed in the Lectures on the dynamical Yang-Baxter equations by P. Etingof and O. Schiffmann, math.QA/9908064. We explicitly compute the matrix entries of the intertwining operator, fusion matrix and exchange matrix associated to sl(2,R) for q=1. We also consider the weighted trace function and the dual Macdonald-Ruijsenaars equation for this particular case. As expected, the matrix entries of the exchange matrix turn out to be Racah polynomials. However, the road to their explicit formula via the fusion matrix is quick, and it also yields an interesting way to derive their orthogonality relations.

math.QA↗
arXiv2000

Some details of proofs of theorems related to the quantum dynamical Yang-Baxter equation

This paper gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in "Lectures on the dynamical Yang-Baxter equation" by P. Etingof and O. Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in "Traces of intertwiners for quantum groups and difference equations, I" by P. Etingof and A. Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations. This paper does not claim originality, priority or completeness. It is meant as a service to whoever may take profit of it.

math.QA↗
arXiv1998

q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations

The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1ψ_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit orthogonal basis for the corresponding L^2-space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big q-Jacobi polynomials. Interpretations on the SU(1,1) and E(2) quantum groups are discussed.

math.CA↗
arXiv1998

Identities of nonterminating series by Zeilberger's algorithm

This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalschütz' summation formula in the non-terminating case this is illustrated.

math.CA↗
arXiv1996

Special functions and q-commuting variables

This paper is mostly a survey, with a few new results. The first part deals with functional equations for q-exponentials, q-binomials and q-logarithms in q-commuting variables and more generally under q-Heisenberg relations. The second part discusses translation invariance of Jackson integrals, q-Fourier transforms and the braided line.

q-alg↗
arXiv1994

Special functions associated with root systems: recent progress

In this paper I present a brief survey of the active area of Special Functions associated with Root Systems. The article is intended for a general mathematical audience. It will not suppose prerequisites on either special functions or root systems. It will also skip many technical details. Some early work in this area (root systems BC_2 and A_2) was done by the author [17] and Sprinkhuizen [32]. During the last ten years important breakthroughs were made by Heckman and Opdam [11], [27], [28], [12], Macdonald [23], [24], Dunkl [8] and Cherednik [3], [4], [5].

math.CA↗
arXiv1994

CQG algebras: a direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.

hep-th↗
arXiv1994

Gauss hypergeometric function and quadratic R-matrix algebras

We consider representations of quadratic $R$-matrix algebras by means of certain first order ordinary differential operators. These operators turn out to act as parameter shifting operators on the Gauss hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with W. Miller's treatment of Lie algebras of first order differential operators will be discussed.

math.QA↗
arXiv1994

General compact quantum groups, a tutorial

A tutorial introduction is given to general Hopf algebras and to general compact quantum groups. In the definition and further treatment of compact quantum groups C*-algebras are avoided. Contact with Woronowicz's compact matrix quantum groups is made at a later stage.

hep-th↗
arXiv1993

Gauss hypergeometric function and quadratic $R$-matrix algebras

We consider representations of quadratic $R$-matrix algebras by means of certain first order ordinary differential operators. These operators turn out to act as parameter shifting operators on the Gauss hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with W. Miller's treatment of Lie algebras of first order differential operators will be discussed.

hep-th↗