A matrix-valued solution to Bochner's problem

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Abstract
We exhibit families of matrix-valued functions F(m,t), m = 0,1,2,...,t real, which are eigenfunctions of a fixed differential operator in t and of a fixed (block) tridiagonal semiinfinite matrix. Thus we have nontrivial solutions of a matrix-valued version of Bochner's problem. These functions arise as matrix-valued spherical functions associated to the two-dimensional complex projective space SU(3)/U(2). In the very special case of one-dimensional representations of U(2) they give instances of Jacobi polynomials that feature among the (scalar-valued) solutions of the problem posed and solved by Bochner back in 1929. This very classical work can be considered as the first instance of the `bispectral problem' of recent interest in several aspects of mathematical physics.

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