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Physical Problems Admitting Heun-to-Hypergeometric Reduction

The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation. Moreover, the translation of the mathematical language for non-mathematicians in making contemporary ideas applied in physical research is not also well developed. In this paper, Maier's Heun-to-hypergeometric reduction cases are studied in detail for four problems in quantum mechanics: (1) The Schrodinger's equation for the Coulomb problem on a 3-sphere, (2) the s-wave bound state equation in the problem of the attractive inverse square potential, (3) the equation for the limit density function for the discrete-time quantum walk, and (4) charged particle under magnetic field and Coulomb force. All these problems give rise to the Heun's equation in their mathematical formulation. A Sage code is given in the appendix for the calculation of some Heun identities.