Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications

@article{Koelink1994AskeyWilsonPA,
  title={Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications},
  author={H. T. Koelink},
  journal={Acta Applicandae Mathematica},
  year={1994},
  volume={44},
  pages={295-352}
}
  • H. Koelink
  • Published 25 July 1994
  • Mathematics
  • Acta Applicandae Mathematica
Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A66 (1990), 146–149] describing the generalised matrix elements in terms of the… 
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References

SHOWING 1-10 OF 51 REFERENCES
Askey-Wilson polynomials as zonal spherical functions on the SU (2) quantum group
On the $SU(2)$ quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of
Orthogonal polynomials in connection with quantum groups
This is a survey of interpretations of q-hypergeometric orthogonal polynomials on quantum groups. The first half of the paper gives general background on Hopf algebras and quantum groups. The
Quantum homogeneous spaces, duality and quantum 2-spheres
For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one
Quantum Groups and q-Orthogonal Polynomials
Many links are already known between quantum groups and q-orthogonal polynomials. In this article, we will give a survey on recent works concerning the realization of q-analogues of the Jacobi
The addition formula for continuous q -Legendre polynomials and associated spherical elements on the SU (2) quantum group related to Askey-Wilson polynomials
The known interpretation of a two-parameter family of Askey–Wilson polynomials as spherical elements on the $SU(2)$ quantum group is extended to an interpretation of a three-parameter (one discrete
Models of Q -algebra representations: the group of plane motions
This paper continues a study of one- and two-variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by
The addition formula for little q-Legendre polynomials and the SU(2) quantum group
From the interpretation of little q-Jacobi polynomials as matrix elements of the irreducible unitary representations of the ${\operatorname{SU}}(2)$ quantum group an addition formula is derived for
...
1
2
3
4
5
...