Jasper V. Stokman

The Askey-Wilson function transform

In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provides an analytical continuation of the Askey-Wilson polynomial in its degree. In this paper we establish the L 2 -theory of the Askey-Wilson function transform, and we explicitly determine its inversion formula.

Koornwinder polynomials and affine Hecke algebras

In this paper we derive the bi-orthogonality relations, diagonal term evaluations and evaluation formulas for the non-symmetric Koornwinder polynomials. For the derivation we use certain representations of the (double) affine Hecke algebra which were originally defined by Noumi and Sahi. The structure of the diagonal terms is clarified by expressing them as residues of the bi-orthogonality weight function. We furthermore give the explicit connection between the non-symmetric and the (anti-)symmetric theory.

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E7 (elliptic, hyperbolic) and of type E6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.

Periodic Integrable Systems with Delta-Potentials

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta-function interactions. The underlying symmetry structures are shown to be governed by the associated graded algebra of Cheredniks (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkins generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.

The Askey-Wilson Function Transform Scheme

In this paper we present an addition to Askey’s scheme of q- hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function. The generalized orthogonality relations and the second order q-differenee equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are discussed shortly.

Trigonometric Cherednik algebra at critical level and quantum many-body problems

Abstract.For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra A(k) at critical level in terms of Dunkl type operators. Under this representation the center of A(k) produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of A(k). We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of A(k).

Koornwinder polynomials and the X X Z spin chain

Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg X X Z spin- 1 2 chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the X X Z spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.

Multivariable big and little q -Jacobi polynomials

A four-parameter family of multivariable big q-Jacobi polynomials and a three-parameter family of multivariable little q-Jacobi polynomials are introduced. For both families, full orthogonality is proved with the help of a second-order q-difference operator which is diagonalized by the multivariable polynomials. A link is made between the orthogonality measures and R. Askeys q-extensions of Selbergs multidimensional beta-integrals.

Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

Solutions to boundary quantum Knizhnik–Zamolodchikov equations are constructed as bilateral sums involving “off-shell” Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of