Matrix-valued orthogonal polynomials related to the quantum analogue of (SU(2)×SU(2),diag)

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of ( SU ( 2 ) × SU ( 2 ) , diag ) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valu...

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Bibliographic Details
Published in:The Ramanujan journal 2017, Vol.43 (2), p.243-311
Main Authors: Aldenhoven, Noud, Koelink, Erik, Román, Pablo
Format: Article
Language:English
Subjects:
Analogs
Chebyshev approximation
Combinatorics
Decomposition
Field Theory and Polynomials
Fourier Analysis
Functions (mathematics)
Functions of a Complex Variable
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix methods
Number Theory
Operators (mathematics)
Orthogonality
Polynomials
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Summary:Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of ( SU ( 2 ) × SU ( 2 ) , diag ) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q -difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q -ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q -ultraspherical polynomials and q -Racah polynomials.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-016-9788-y