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

N. ALDENHOVEN; E. KOELINK; P. ROMÁN

RAMANUJAN JOURNAL

SPRINGER

Lugar: Berlin; Año: 2016 vol. 43 p. 243 - 311

1382-4090

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) imes SU(2), diag)$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal envelopingalgebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analoguesof 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 fromthe Schur orthogonality relations, the three-term recurrence relationand the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, 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.