Matrix-valued orthogonal polynomials related to the quantum analogue of
arXiv:1507.03426 · doi:10.1007/s11139-016-9788-y
Abstract
Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of 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, the matrix-valued Askey-Wilson type -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 -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 -ultraspherical polynomials and -Racah polynomials.
42 pages, accompanying sage-file available through author's website. V2: correction of Remark 6.5 with proof, references updated