Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

@article{Koelink1996ConvolutionsFO,
  title={Convolutions for orthogonal polynomials from Lie and quantum algebra representations.},
  author={H. T. Koelink and Joris Van der Jeugt},
  journal={Siam Journal on Mathematical Analysis},
  year={1996},
  volume={29},
  pages={794-822}
}
Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra ${\frak{su}}(1,1)$ and the Clebsch--Gordan decomposition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the quantized universal enveloping algebra for ${\frak{su}}(1,1… 
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References

SHOWING 1-10 OF 42 REFERENCES
Spectral theory of Jacobi matrices in l 2 ( Z ) and the su (1,1) lie algebra
The connection between orthogonal polynomials, continued fractions, difference equations, and self-adjoint Jacobi matrices acting in $l^2 (\mathbb{Z}^ + )$ and the extension of these connections to
Models of q-algebra representations: Tensor products of special unitary and oscillator algebras
This paper begins 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
Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications
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
A difference equation and Hahn polynomials in two variables.
The space of polynomials in N variables spanned by squarefree monomials of degree r and annihilated by Σ?=ιd/dXi furnishes an irreducible representation of SN, the symmetric group on N objects. The
Spectra, eigenvectors and overlap functions for representation operators ofq-deformed algebras
Operators of representations corresponding to symmetric elements of theq-deformed algebrasUq(su1,1),Uq(so2,1),Uq(so3,1),Uq(son) and representable by Jacobi matrices are studied. Closures of unbounded
Coupling coefficients for Lie algebra representations and addition formulas for special functions
Representations of the Lie algebra su(1,1) and of a generalization of the oscillator algebra, b(1), are considered. The paper then introduces polynomials which are related by the coupling (or
Orthogonal Polynomials in Two Variables of q-Hahn and q-Jacobi Type
  • C. Dunkl
  • Computer Science, Mathematics
    SIAM J. Algebraic Discret. Methods
  • 1980
TLDR
Two families of orthogonal polynomials in two discrete variables are constructed for a weight function of q-hypergeometric type, and corresponding results are obtained for Andrews and Askey’s little q-Jacobi polynmials, which are Orthogonal on a countable compact set.
Convolutions of Orthonormal Polynomials
In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad
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
On Jacobi and continuous Hahn polynomials
Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality
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