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Many special functions are solutions of both, a differential and a functional equation. We use this duality to solve a large class of abstract Sturm-Liouville equations, initiating a theory of Sturm-Liouville operator functions; cosine, Bessel, and Legendre operator functions are contained as special cases. This is part of a general concept of operator functions being multiplicative with respect to convolution of a hypergroupcontaining all representations of (hyper)groups, and further abstractdoi:10.4064/dm763-3-2018 fatcat:ms7nqm6vtvgtrepoujm2zpzlr4