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}
}
• Published 9 July 1996
• Mathematics
• Siam Journal on Mathematical Analysis
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 ${\frak{su}}(1,1… Convolution identities for Dunkl orthogonal polynomials from the osp(1|2) Lie superalgebra • Mathematics, Physics Journal of Mathematical Physics • 2019 New convolution identities for orthogonal polynomials belonging to the q = −1 analog of the Askey-scheme are obtained. Specialization of the Chihara polynomials will play a central role as the Wilson Function Transforms Related to Racah Coefficients The irreducible$*$-representations of the Lie algebra $${\mathfrak {\rm u}}(1,1)$$consist of discrete series representations, principal unitary series and complementary series. We calculate Racah Bilinear Summation Formulas from Quantum Algebra Representations The tensor product of a positive and a negative discrete series representation of the quantum algebra Uq(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Bilinear Generating Functions for Orthogonal Polynomials • Mathematics • 1997 Abstract. Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these Heun algebras of Lie type • Mathematics Proceedings of the American Mathematical Society • 2019 We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For$\mathfrak{su}(2)$, this leads to the A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials • W. Groenevelt • Mathematics International Mathematics Research Notices • 2019 We study matrix elements of a change of basis between two different bases of representations of the quantum algebra${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are Hamiltonian Type Operators in Representations of the Quantum Algebra suq(1,1) • Mathematics • 2004 We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of Big q-Laguerre and q-Meixner polynomials and representations of the quantum algebra Uq(su1,1) • Mathematics, Physics • 2003 Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are BILINEAR GENERATING FUNCTIONSFOR ORTHOGONAL POLYNOMIALS • 1997 Using realisations of the positive discrete series representations of the Lie algebra su(1; 1) in terms of Meixner-Pollaczek polynomials, the action of su(1; 1) on Poisson kernels of these Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions • Mathematics Symmetry, Integrability and Geometry: Methods and Applications • 2018 Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its$q$-analogue. The resulting References SHOWING 1-10 OF 42 REFERENCES Spectral theory of Jacobi matrices in l 2 ( Z ) and the su (1,1) lie algebra • Mathematics • 1991 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 • Mathematics • 1992 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 • Mathematics • 1996 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 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 • Mathematics • 1976 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