Three Term Recursion Relation for Spherical Functions Associated to the Complex Projective Plane

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The aim of this paper is to prove a three term recursion relation for a sequence of matrix valued functions \(\widetilde\Phi\)(g,w) on G=SU(3) built up of l+1 spherical functions of a given type πn,l, associated to the complex projective plane G/K, K=S(U(2)×U(1)). The three term recursion relation that constitutes our main result, Theorem 5.2, together with the fact that the functions \(\widetilde\Phi\)(g,w) are eigenfunctions of all differential operators on G which are left invariant under G and right invariant under K, provides for each l∈N0 a solution of a matrix valued extension of the Bochner's problem to G. In fact by restriction to an Abelian Iwasawa subgroup of G, for each l∈N0, we obtain a sequence \({\widetilde H}\)(t,w) of matrix valued polynomial functions on t which satisfies a three term recursion relation and such that they are eigenfunctions of a second order differential operator on 02<t<1. Thus each sequence \({\widetilde H}\)(t,w) satisfies both conditions explicitly asked for by Bochner.