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

Abstract

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.