### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 071, 41 pages      arXiv:1312.0909      https://doi.org/10.3842/SIGMA.2014.071

### Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere

Juan Alfredo Tirao and Ignacio Nahuel Zurrián
CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina

Received December 20, 2013, in final form June 20, 2014; Published online July 07, 2014

Abstract
In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=({\mathrm{SO}}(n+1),{\mathrm{SO}}(n))$ in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same $K$-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$ supported on $[0,1]$. Moreover, we show that $W$ has a second order symmetric hypergeometric operator $D$.

Key words: matrix-valued spherical functions; matrix orthogonal polynomials; the matrix hypergeometric operator; $n$-dimensional sphere.

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