This article is cited in 9 papers

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

Juan Alfredo Tirao, Ignacio Nahuel Zurrián

CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina

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 $D$.

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

MSC: 22E45; 33C45; 33C47

Received: December 20, 2013; in final form June 20, 2014; Published online July 7, 2014

Language: English

DOI: 10.3842/SIGMA.2014.071

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ArXiv: 1312.0909