Matrix valued orthogonal polynomials related to (SU(2)×SU(2),diag)

E. KOELINK; M. VAN PRUIJSSEN; P. ROMÁN

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2012 vol. 2012 p. 5673 - 5730

1073-7928

The matrix-valued spherical functions for the pair (K × K, K), K = SU(2), are studied. By restriction to the subgroup A, the matrix-valued spherical functions are diagonal. For suitable set of spherical functions, we take these diagonals as a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions, we obtain matrix- valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression, we obtain right- hand-sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues? formula and being eigenfunctions to first-order differential?difference and second-order differential operators, are obtained.